2 The Mathematical Foundations of Statistical Experiments
2.1 The Sample Space and Events
In Chapter 1 we saw the mathematical formulation for the Frequentist interpretation of probability. To study probability, we require a more detailed mathematical model. We want a description, framed in terms of mathematical objects, which will allow us to work out probabilities of interest. In general, to form such a probability model we need both a list of all possible outcomes that the experiment can produce, as well as the probabilities of these outcomes.
We call the list of outcomes that can occur from an experiment the sample space of the experiment. The sample space is denoted \(\mathcal{S}\), and is defined as the set of all possible outcomes from the experiment. For instance, if the experiment is flipping a coin we have \(\mathcal{S} = \{\text{H}, \text{T}\}\). If the experiment is rolling a six-sided die then \(\mathcal{S} = \{1,2,3,4,5,6\}\).
Definition 2.1 (Sample Space) The sample space of a statistical experiment is the set of all possible outcomes that can be realized from that experiment. The sample space is typically denoted \(\mathcal{S}\), or with similar script letters.
Example 2.1 (Enumerating Sample Spaces) Write down the complete sample space, \(\mathcal{S}\) for the game that Sadie and Charles play, based on flipping and observing a coin three times in sequence.
With the sample space formally defined, we can revisit Definition 1.2, and formally define the concept of an event.
Definition 2.2 (Event) An event is any collection of outcomes from a sample space for a statistical experiment. Mathematically, an event, \(E\), is a subset of \(\mathcal{S}\), and we write \(E\subset\mathcal{S}\).
Take for instance the experiment of a single coin. In this case, we may define \(E_1 = \{\text{H}\}\), \(E_2 = \{\text{T}\}\), and \(E_3 = \{\text{H},\text{T}\}\) as examples of possible events. Here, \(E_1\) corresponds to the event that a head is observed, \(E_2\) corresponds to the event that a tail is observed, and \(E_3\) corresponds to the event that either a tails or a heads was observed. Note that for each event we have \(E_1 \subseteq\mathcal{S}\), \(E_2 \subseteq\mathcal{S}\), and \(E_3 \subseteq \mathcal{S}\).1
Example 2.2 (Basic Event Listing) List several events from the game that Charles and Sadie are playing. Indicate why these are events.
Example 2.3 (Event Identification) Is “Charles has to pay” an event from the game that Charles and Sadie are playing? Why? Explain.
Example 2.4 (Defining Events from Real-World Descriptions) What event corresponds to the description “Sadie has to pay” in the game that Charles and Sadie are playing? Recall that they flip a coin three times, and Charles will pay if at least two heads come up, while Sadie will pay if at least two tails come up.
While the events \(E_1 = \{\text{H}\}\) and \(E_2 = \{\text{T}\}\) each correspond to a simple outcome from the sample space, \(E_3 = \{\text{H},\text{T}\}\) corresponds to a combined event. We call direct outcomes simple events and more complex outcomes like \(E_3\) compound events.
Definition 2.3 (Simple Event) A simple event is any event which corresponds to exactly one outcome from the sample space. A simple event only has one way of occurring. The size of the set for a simple event will be \(1\). The sample space, in turn, is made up of a collection of simple events.
Definition 2.4 (Compound Event) A compound event is any event which corresponds to more than one outcome from the sample space. A compound event can occur in multiple different ways. The size of the set for the compound event will be greater than \(1\).
If we consider rolling a six-sided die, then an example of a simple event is that a four shows up, denoted \(\{4\}\). A compound event could be that an even number is rolled, \(\{2,4,6\}\), or that a number greater than or equal to four is rolled, \(\{4, 5, 6\}\).
Example 2.5 (Identifying Simple and Compound Events) List an example of (at least) one simple and one compound event from the game that Charles and Sadie are playing.
We say that an event “occurs” if any of the outcomes comprising the event occur. As a result we can have more than one event occurring as the result of a run of a statistical experiment. Suppose that we are rolling a fair, six-sided die. Consider the events “an even number was rolled” and “a number greater than or equal to four was rolled.” If a four or a six are rolled, both of these events happen simultaneously. Our goal when working with probability will be to assign probability values to different events. We will talk about how likely, or unlikely, events of interest are, given the underlying statistical experiment.
Above, we defined \(E_3\) to be equal to \(\mathcal{S}\). As a result, we can say that \(\mathcal{S}\) is an event since \(\mathcal{S} \subseteq \mathcal{S}\). This is the event that any outcome is observed, which is certain to happen. Since it is certain to happen, we know it happens with probability \(1\). There is another “special” event which is important to consider. We call this the null event. Denoted \(\emptyset\), the null event is an event that corresponds to “nothing in the sample space”. We know that every time an experiment is run something in the sample space occurs, and so the null event is assigned probability zero.
Definition 2.5 (Null Event) The null event, denoted \(\emptyset\) or \(\{\}\), is an event from a statistical experiment which corresponds to nothing within the sample space. The null event has probability zero, and it is impossible to observe. Note that, no matter the sample space, \(\emptyset\subset\mathcal{S}\).
2.2 Set Operations for Event Manipulation
Ultimately, all events are sets. These sets are subsets of the sample space, and can contain single or multiple outcomes. Every quantity that we are interested in can be expressed as some set of outcomes of interest. In building up these sets it is common to construct them through the use of “and”, “or”, and “not” statements. That is, we may say that our event occurs if some outcome OR another outcome occurs, or perhaps our outcomes occurs if some outcome does NOT occur.3
Consider the example of drawing cards from a standard 52-card deck. In such a deck there are 13 card ranks, and four card suits, with one of each combination present. If we draw a single card we can think of the outcomes of the experiment as being any of the 52 possible combinations of rank and suit. We are often interested in an event such as “the card is red”, which is the same as saying “the card is a heart or the card is a diamond.” Perhaps we want to know whether the card was an ace through ten, this is the same as saying “the card is not a Jack or a Queen or a King.” If we are interested in the event that the ace of spades was drawn, this can be expressed by saying that “the card was a spade and the card was an ace.”
As you begin to pay attention to the linguistic representation of events that we use, you will notice more and more the use of these words to form compound events in particular. As a result, we give each of them a mathematical operation which allow us to quickly and compactly express these quantities in notation.
Example 2.6 (Description of Events) Describe “Charles has to pay”, based on the game Charles and Sadie are playing, using language revolving around “or”, “and”, and “not” each. That is, describe observing at least two heads, on three flips of a coin, one time using “or”, one time using “and”, and one time using “not”.4
We define mathematical operations to encapsulate the use of or, and, and not. These operations apply to any mathematical sets, whether they refer to events or not.
Definition 2.6 (Union) The union encodes the use of “or” in reference to two or more sets. Formally, with two sets \(A\) and \(B\), the union of \(A\) and \(B\) is the set of all elements that are contained in \(A\), or \(B\), or both \(A\) and \(B\). We write \(A \cup B\) and read that as \(A\) union \(B\). When we wish to take the union of many sets, \(A_1,A_2,\dots,A_n\), we write this as \[A_1 \cup A_2 \cup \cdots \cup A_n = \bigcup_{i=1}^n A_i.\]
Definition 2.7 (Intersection) The intersection captures the use of “and” in reference to two or more sets. Formally, the intersection of two sets, \(A\), and \(B\), is the set that contains all elements that belong to both \(A\) and \(B\). We write \(A \cap B\), and say “\(A\) intersect \(B\).” When we wish to take the intersection of many sets, \(A_1,A_2,\dots,A_n\), we write \[A_1 \cap A_2 \cap \cdots \cap A_n = \bigcap_{i=1}^n A_i.\]
Definition 2.8 (Complement) The complement makes formal the concept of “not.” The complement of a set is the set of all elements which occur in the sample space but are not in the given set. We write this as \(A^C\) and say “\(A\) complement.” When dealing with a sample space, \(\mathcal{S}\), the complement of \(A\) is the set of all elements in \(\mathcal{S}\) that are not in \(A\).
Example 2.7 (Basic Set Operations) For the game being played by Charles and Sadie, take \(E_1 = \{(\text{H}, \text{H}, \text{H})\}\), \(E_2 = \{(\text{H}, \text{H}, \text{H}), (\text{T}, \text{H}, \text{H}), (\text{H}, \text{T}, \text{H})\}\), and \(E_3 = \{(\text{H}, \text{H}, \text{T})\}\). Express the following events.
- \(E_1 \cup E_2\);
- \(E_1 \cap E_2\);
- \(E_2^C\);
- \(E_2 \cap E_3\);
- \(E_1 \cup E_2 \cup E_3\).
Definition 2.9 (Disjoint Events) Two events, \(E_1\) and \(E_2\) are said to be disjoint whenever their intersection is the null event. That is, if \(E_1 \cap E_2 = \emptyset\) then \(E_1\) and \(E_2\) are disjoint events.
These concepts allow us to more compactly express sets of interest, and in particular, will be quite useful when it comes to assigning probability. The more times you work with the set operations, the more familiar they will become, and as a result, practice is always useful.
Consider rolling a 6-sided die, and take \(A\) to be the event that a \(6\) is rolled, \(B\) to be the event that the roll was at least \(5\), \(C\) to be the event that the roll was less than \(4\), and \(D\) to be the event that the roll was odd.
- If we consider \(D^C\) this is the event that the roll was even;
- \(A \cup C\) is the event that a \(6\) was rolled or that a number less than \(4\) was rolled, which is to say anything other than a \(4\) or a \(5\), which we may also express as \(\{4,5\}^C\).
- If we take \(A \cup B\) then this will be the same as \(B\), and \(A\cap B\) will be \(A\).
- If we take the event \(A\cap C\), notice that no outcomes satisfy both conditions, and so \(A \cap C = \emptyset\).
- We can also join together multiple operations. \(D^C \cap C\) gives us even numbers than less than \(4\), which is to say the outcome \(2\).
- Similarly, \((A \cap B)^C\) would represent the event that a number less than \(6\) is rolled.
Example 2.8 (Set Operations with Decks of Cards) Charles and Sadie are tiring of flipping their coin, and so they wish to start using decks of cards sometimes instead. Before they formalize a game based on decks of cards, they want to make sure that they are both very comfortable working with these. Suppose that the sample space is defined to be the set of \(52\) standard cards that may be drawn on a single draw from the deck. Describe how set operations can be used to form events corresponding to:
- A red card is observed.
- Any card between an ace and a ten is observed.
- The ace of spades is observed.
Working with these basic set operations should eventually become second nature. There are often very many ways of expressing the same event using these different operations, and finding the most useful method of representing a particular event can often be the key to solving challenging probability questions. The first step in making sure that these tools are available to you is in ensuring that the basic operations are fully understood, and this comes via practice. Remember, unions represent “ors”, intersections represent “ands”, and complements represent “nots”.
2.2.1 Using R To Represent Sample Spaces and Events and Performing Set Operations
We have seen how R can encode sets of elements using vectors. For instance, we may take sample_space <- 1:6 to represent the sample space of rolling a six-sided die. We can form events by taking subsets of the relevant quantities, selecting via indices. Fortunately, there are also all of the basic set operations implemented in R.
We can use union(x, y) to perform the union of x and y, intersect(x, y) to perform the intersection of x and y, and setdiff(x = sample_space, y) to perform the complement of y (assuming that sample_space contains the full sample space).9
2.3 Venn Diagrams
The sample space is partitioned into outcomes, and the outcomes can be grouped together into events. These events are sets and can be manipulated via basic set operations. Sometimes it is convenient to represent this process graphically through the use of Venn diagrams. In a Venn diagram, the sample space is represented by a rectangle with the possible outcomes placed inside, and events are drawn inside of this as circles containing the relevant outcomes.
On the Venn diagram then, the overlap between circles represents their intersection, the combined area of two (or more) circles represents their union, and everything outside of a given circle represents the complement. This can be a fairly useful method for representing sample spaces, and for visualizing the basic set operations that we use to manipulate events inside the sample spaces.
A word of caution: Venn diagrams are useful tools, but they are not suitable as mathematical proofs directly. It is possible to convince yourself of false truths if the wrong diagrams are used, and as a result, Venn diagrams should be thought of as aids to understanding, rather than as a rigorous tool in and of themselves.10
Example 2.9 (Venn Diagram with Defined Events) Draw a Venn diagram representing the original game that Charles and Sadie played. On the diagram draw the events corresponding to “At least one head and one tail are observed”, and “Sadie won the game”. Recall that three coins are tossed, and Sadie wins if at least two of them show heads.
The sample space contains the eight possible options. Only \((\text{T}, \text{T}, \text{T})\) does not belong to at least one of the events. Both events share \((\text{H},\text{H},\text{T})\), \((\text{H},\text{T},\text{H})\), and \((\text{T},\text{H},\text{H})\).
Sample spaces, events, and the manipulation of these quantities forms a critical component of understanding probability models. In particular, they describe the complete set of occurrences in a statistical experiment that we could be interested in assigning probability values to. To formalize a probability model, however, we also need some rule for assigning probability values.
Self-Assessment
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Every time an experiment is run, exactly one event occurs.
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The probability that an event occurs is always 0.5, since it either occurs or does not.
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The sample space is the set containing all possible events.
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Events are described using mathematical sets.
(Question ID: 0870013095)
The sample space is the set containing all possible events.
(Question ID: 0030580179)
The sample space is the set containing all possible events.
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Events are described using mathematical sets.
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An event can contain more than one possible outcome.
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It is possible for an event to contain no outcomes.
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The probability of an event is a number between 0 and 1, inclusive.
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It is possible for an event to contain no outcomes.
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An event can contain more than one possible outcome.
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It is possible for two events to occur from one run of a random experiment.
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If an event occurs, then the probability of the event is equal to 1.
(Question ID: 0978435949)
The sample space is the set containing all possible events.
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The sample space of an experiment can be considered an event.
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If an event occurs, then the probability of the event is equal to 1.
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An event can contain more than one possible outcome.
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Events are the outcomes of an experiment.
(Question ID: 0687549664)
Every time an experiment is run, exactly one event occurs.
(Question ID: 0800045695)
If an event occurs, then the probability of the event is equal to 1.
(Question ID: 0047453828)
The probability that an event occurs is always 0.5, since it either occurs or does not.
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Events are the outcomes of an experiment.
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If an event did not occur, then the probability of the event is 0.
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Events are the outcomes of an experiment.
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Events are defined to be any subset of the sample space.
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It is possible to observe an event with elements that are outside the sample space.
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It is possible to observe an event with elements that are outside the sample space.
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A sample space must be a finite set.
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We typically denote the sample space using \(\mathcal{S}\), or similar.
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The sample space is the set of all possible events in a statistical experiment.
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The sample space is the set of all possible events in a statistical experiment.
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Events are defined to be any subset of the sample space.
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The sample space of a statistical experiment contains all possible outcomes.
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A sample space must be a finite set.
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The sample space contains possible outcomes and their associated probabilities.
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The sample space contains possible outcomes and their associated probabilities.
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We typically denote the sample space using \(\mathcal{S}\), or similar.
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The sample space can be thought of as an event from the random experiment.
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There can be repeated or overlapping elements in a sample space.
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For each of the following, identify whether the event is simple or compound.
- Rolling a 6 on a six-sided die.
- No tails are seen when flipping a coin 23 times.
- Only tails are seen when flipping a coin 23 times.
(Question ID: 0741213616)
For each of the following, identify whether the event is simple or compound.
- At least one tails is seen when flipping a coin 26 times.
- Only tails are seen when flipping a coin 26 times.
- Rolling a number less than 5 on a six-sided die.
(Question ID: 0581104406)
For each of the following, identify whether the event is simple or compound.
- The same number is seen when rolling a six-sided die 38 times.
- Drawing the Ace of Spades from a standard deck of cards.
- At least one tails is seen when flipping a coin 38 times.
(Question ID: 0414665215)
For each of the following, identify whether the event is simple or compound.
- Only tails are seen when flipping a coin 50 times.
- Drawing the Ace of Spades from a standard deck of cards.
- Picking a red marble from a bag containing 38 red and 16 black marbles.
(Question ID: 0895720886)
For each of the following, identify whether the event is simple or compound.
- An odd number is seen when rolling a six-sided die.
- At least one tails is seen when flipping a coin 81 times.
- Rolling a 4 on a six-sided die.
(Question ID: 0203039819)
For each of the following, identify whether the event is simple or compound.
- The same result is seen when flipping a coin 99 times.
- Rolling a number less than 6 on a six-sided die.
- A red card is drawn from a standard deck of cards.
(Question ID: 0842052695)
For each of the following, identify whether the event is simple or compound.
- Drawing the Ace of Spades from a standard deck of cards.
- Picking a red marble from a bag containing 4 red and 47 black marbles.
- A black card is drawn from a standard deck of cards.
(Question ID: 0412403743)
For each of the following, identify whether the event is simple or compound.
- The same number is seen when rolling a six-sided die 55 times.
- The same result is seen when flipping a coin 55 times.
- Rolling a 6 on a six-sided die.
(Question ID: 0074817175)
For each of the following, identify whether the event is simple or compound.
- Picking a red marble from a bag containing 14 red and 1 black marbles.
- Only tails are seen when flipping a coin 69 times.
- No tails are seen when flipping a coin 69 times.
(Question ID: 0324225617)
For each of the following, identify whether the event is simple or compound.
- At least one heads is seen when flipping a coin 88 times.
- The same result is seen when flipping a coin 88 times.
- The same number is seen when rolling a six-sided die 88 times.
(Question ID: 0001361110)
For each of the following, identify whether the event is simple or compound.
- A red card is drawn from a standard deck of cards.
- At least one tails is seen when flipping a coin 14 times.
- The same number is seen when rolling a six-sided die 14 times.
(Question ID: 0509376007)
For each of the following, identify whether the event is simple or compound.
- Rolling a 1 on a six-sided die.
- Rolling a number less than 4 on a six-sided die.
- Drawing the Ace of Spades from a standard deck of cards.
(Question ID: 0727958617)
For each of the following, identify whether the event is simple or compound.
- Rolling a number less than 5 on a six-sided die.
- At least one tails is seen when flipping a coin 10 times.
- Only tails are seen when flipping a coin 10 times.
(Question ID: 0792206038)
For each of the following, identify whether the event is simple or compound.
- The same result is seen when flipping a coin 81 times.
- Rolling a 3 on a six-sided die.
- Only heads are seen when flipping a coin 81 times.
(Question ID: 0038517045)
For each of the following, identify whether the event is simple or compound.
- Rolling a number less than 4 on a six-sided die.
- Flipping a coin and getting tails.
- Only tails are seen when flipping a coin 43 times.
(Question ID: 0250708588)
For each of the following, identify whether the event is simple or compound.
- A red card is drawn from a standard deck of cards.
- Only heads are seen when flipping a coin 75 times.
- Flipping a coin and getting heads.
(Question ID: 0794606368)
For each of the following, identify whether the event is simple or compound.
- Picking a red marble from a bag containing 18 red and 49 black marbles.
- Rolling a 4 on a six-sided die.
- A black card is drawn from a standard deck of cards.
(Question ID: 0105950044)
For each of the following, identify whether the event is simple or compound.
- Picking a red marble from a bag containing 49 red and 28 black marbles.
- Rolling a 4 on a six-sided die.
- Only tails are seen when flipping a coin 14 times.
(Question ID: 0096299433)
For each of the following, identify whether the event is simple or compound.
- Picking a red marble from a bag containing 8 red and 31 black marbles.
- A black card is drawn from a standard deck of cards.
- Flipping a coin and getting tails.
(Question ID: 0750412833)
For each of the following, identify whether the event is simple or compound.
- Drawing the Ace of Spades from a standard deck of cards.
- The same number is seen when rolling a six-sided die 88 times.
- No tails are seen when flipping a coin 88 times.
(Question ID: 0088516848)
For each of the following, identify whether the event is simple or compound.
- Rolling a 5 on a six-sided die.
- Picking a red marble from a bag containing 38 red and 25 black marbles.
- The same number is seen when rolling a six-sided die 70 times.
(Question ID: 0173770106)
For each of the following, identify whether the event is simple or compound.
- Picking a red marble from a bag containing 37 red and 45 black marbles.
- Drawing the Ace of Spades from a standard deck of cards.
- The same number is seen when rolling a six-sided die 54 times.
(Question ID: 0763041990)
For each of the following, identify whether the event is simple or compound.
- Drawing the Ace of Spades from a standard deck of cards.
- An odd number is seen when rolling a six-sided die.
- Rolling a number less than 3 on a six-sided die.
(Question ID: 0986174424)
For each of the following, identify whether the event is simple or compound.
- The same number is seen when rolling a six-sided die 52 times.
- The same result is seen when flipping a coin 52 times.
- Only tails are seen when flipping a coin 52 times.
(Question ID: 0257867851)
For each of the following, identify whether the event is simple or compound.
- No heads are seen when flipping a coin 69 times.
- Flipping a coin and getting heads.
- Rolling a number less than 5 on a six-sided die.
(Question ID: 0310581232)
For each of the following, identify whether the event is simple or compound.
- Drawing the Ace of Spades from a standard deck of cards.
- A black card is drawn from a standard deck of cards.
- Rolling a number less than 6 on a six-sided die.
(Question ID: 0049150372)
For each of the following, identify whether the event is simple or compound.
- Drawing the Ace of Spades from a standard deck of cards.
- Rolling a 5 on a six-sided die.
- Picking a red marble from a bag containing 2 red and 2 black marbles.
(Question ID: 0066682472)
For each of the following, identify whether the event is simple or compound.
- Rolling a number less than 3 on a six-sided die.
- Rolling a 1 on a six-sided die.
- The same result is seen when flipping a coin 68 times.
(Question ID: 0763729930)
For each of the following, identify whether the event is simple or compound.
- At least one heads is seen when flipping a coin 17 times.
- No heads are seen when flipping a coin 17 times.
- Flipping a coin and getting heads.
(Question ID: 0265420981)
For each of the following, identify whether the event is simple or compound.
- Only tails are seen when flipping a coin 62 times.
- A red card is drawn from a standard deck of cards.
- At least one tails is seen when flipping a coin 62 times.
(Question ID: 0693850884)
For each of the following, identify whether the event is simple or compound.
- Only heads are seen when flipping a coin 95 times.
- A red card is drawn from a standard deck of cards.
- The same number is seen when rolling a six-sided die 95 times.
(Question ID: 0555590573)
For each of the following, identify whether the event is simple or compound.
- Rolling a 4 on a six-sided die.
- Flipping a coin and getting heads.
- No heads are seen when flipping a coin 97 times.
(Question ID: 0605786154)
For each of the following, identify whether the event is simple or compound.
- The same number is seen when rolling a six-sided die 48 times.
- At least one heads is seen when flipping a coin 48 times.
- No heads are seen when flipping a coin 48 times.
(Question ID: 0309781015)
For each of the following, identify whether the event is simple or compound.
- The same result is seen when flipping a coin 72 times.
- A red card is drawn from a standard deck of cards.
- Only heads are seen when flipping a coin 72 times.
(Question ID: 0534512275)
For each of the following, identify whether the event is simple or compound.
- Rolling a number less than 3 on a six-sided die.
- The same number is seen when rolling a six-sided die 51 times.
- No heads are seen when flipping a coin 51 times.
(Question ID: 0382806640)
For each of the following, identify whether the event is simple or compound.
- Rolling a number less than 5 on a six-sided die.
- An odd number is seen when rolling a six-sided die.
- A red card is drawn from a standard deck of cards.
(Question ID: 0795829595)
For each of the following, identify whether the event is simple or compound.
- The same number is seen when rolling a six-sided die 90 times.
- Drawing the Ace of Spades from a standard deck of cards.
- A red card is drawn from a standard deck of cards.
(Question ID: 0173760205)
For each of the following, identify whether the event is simple or compound.
- An even number is seen when rolling a six-sided die.
- Only heads are seen when flipping a coin 73 times.
- The same number is seen when rolling a six-sided die 73 times.
(Question ID: 0635141237)
For each of the following, identify whether the event is simple or compound.
- Only heads are seen when flipping a coin 50 times.
- Drawing the Ace of Spades from a standard deck of cards.
- Rolling a 3 on a six-sided die.
(Question ID: 0341121683)
For each of the following, identify whether the event is simple or compound.
- Flipping a coin and getting heads.
- Rolling a number less than 4 on a six-sided die.
- No heads are seen when flipping a coin 47 times.
(Question ID: 0319575028)
For each of the following, identify whether the event is simple or compound.
- Rolling a number less than 4 on a six-sided die.
- The same result is seen when flipping a coin 59 times.
- No heads are seen when flipping a coin 59 times.
(Question ID: 0906187433)
For each of the following, identify whether the event is simple or compound.
- Rolling a 6 on a six-sided die.
- Drawing the Ace of Spades from a standard deck of cards.
- The same number is seen when rolling a six-sided die 59 times.
(Question ID: 0534036966)
For each of the following, identify whether the event is simple or compound.
- Drawing the Ace of Spades from a standard deck of cards.
- At least one heads is seen when flipping a coin 61 times.
- The same number is seen when rolling a six-sided die 61 times.
(Question ID: 0975755816)
For each of the following, identify whether the event is simple or compound.
- Rolling a 6 on a six-sided die.
- The same result is seen when flipping a coin 12 times.
- Drawing the Ace of Spades from a standard deck of cards.
(Question ID: 0818960261)
For each of the following, identify whether the event is simple or compound.
- Flipping a coin and getting heads.
- An even number is seen when rolling a six-sided die.
- Rolling a 3 on a six-sided die.
(Question ID: 0921075178)
For each of the following, identify whether the event is simple or compound.
- At least one tails is seen when flipping a coin 42 times.
- A black card is drawn from a standard deck of cards.
- No tails are seen when flipping a coin 42 times.
(Question ID: 0038045636)
For each of the following, identify whether the event is simple or compound.
- Flipping a coin and getting heads.
- Rolling a 6 on a six-sided die.
- The same number is seen when rolling a six-sided die 79 times.
(Question ID: 0116870503)
For each of the following, identify whether the event is simple or compound.
- A red card is drawn from a standard deck of cards.
- No heads are seen when flipping a coin 55 times.
- Rolling a 2 on a six-sided die.
(Question ID: 0918460167)
For each of the following, identify whether the event is simple or compound.
- Flipping a coin and getting tails.
- An odd number is seen when rolling a six-sided die.
- The same number is seen when rolling a six-sided die 32 times.
(Question ID: 0844654565)
For each of the following, identify whether the event is simple or compound.
- Picking a red marble from a bag containing 20 red and 20 black marbles.
- The same number is seen when rolling a six-sided die 17 times.
- Drawing the Ace of Spades from a standard deck of cards.
(Question ID: 0356064475)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}\). Moreover, take three events, \(A = \{8, 11, 12, 14\}\), \(B = \{8, 11, 12, 15\}\), and \(C = \{11, 14, 16, 17\}\).
Which of the following elements comprise \(B\cup C^C\)? (Note, select all that apply).
(Question ID: 0830789004)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}\). Moreover, take three events, \(A = \{6, 9, 10, 11\}\), \(B = \{4, 7, 8, 10\}\), and \(C = \{3, 5, 9, 11\}\).
Which of the following elements comprise \(B^C\cap C\)? (Note, select all that apply).
(Question ID: 0755796523)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\). Moreover, take three events, \(A = \{12, 14, 16, 18\}\), \(B = \{11, 12, 17, 18\}\), and \(C = \{14, 15, 18, 19\}\).
Which of the following elements comprise \(B\cup C^C\)? (Note, select all that apply).
(Question ID: 0974255443)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Moreover, take three events, \(A = \{0, 2, 6, 9\}\), \(B = \{0, 5, 6, 8\}\), and \(C = \{2, 5, 6, 8\}\).
Which of the following elements comprise \(A\cap B\)? (Note, select all that apply).
(Question ID: 0717782568)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}\). Moreover, take three events, \(A = \{5, 10, 12, 13\}\), \(B = \{5, 7, 8, 9\}\), and \(C = \{4, 5, 7, 12\}\).
Which of the following elements comprise \(A\cap B\)? (Note, select all that apply).
(Question ID: 0854207875)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}\). Moreover, take three events, \(A = \{2, 4, 5, 6\}\), \(B = \{7, 8, 10, 11\}\), and \(C = \{5, 6, 8, 11\}\).
Which of the following elements comprise \(A^C\cap B\)? (Note, select all that apply).
(Question ID: 0991274741)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{9, 10, 11, 12, 13, 14, 15, 16, 17, 18\}\). Moreover, take three events, \(A = \{9, 11, 13, 18\}\), \(B = \{10, 12, 13, 16\}\), and \(C = \{12, 14, 15, 17\}\).
Which of the following elements comprise \(A\cup B^C\)? (Note, select all that apply).
(Question ID: 0554537473)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{9, 10, 11, 12, 13, 14, 15, 16, 17, 18\}\). Moreover, take three events, \(A = \{11, 14, 15, 18\}\), \(B = \{10, 13, 16, 17\}\), and \(C = \{9, 12, 14, 15\}\).
Which of the following elements comprise \(B^C\cup C\)? (Note, select all that apply).
(Question ID: 0386082712)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\). Moreover, take three events, \(A = \{11, 12, 13, 19\}\), \(B = \{13, 15, 16, 17\}\), and \(C = \{11, 14, 15, 19\}\).
Which of the following elements comprise \(A\cup B\)? (Note, select all that apply).
(Question ID: 0308784659)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{5, 6, 7, 8, 9, 10, 11, 12, 13, 14\}\). Moreover, take three events, \(A = \{7, 8, 12, 14\}\), \(B = \{8, 10, 11, 12\}\), and \(C = \{7, 10, 12, 13\}\).
Which of the following elements comprise \(A\cap B^C\)? (Note, select all that apply).
(Question ID: 0589371863)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{9, 10, 11, 12, 13, 14, 15, 16, 17, 18\}\). Moreover, take three events, \(A = \{9, 14, 15, 16\}\), \(B = \{11, 15, 16, 17\}\), and \(C = \{9, 16, 17, 18\}\).
Which of the following elements comprise \(B^C\cap C\)? (Note, select all that apply).
(Question ID: 0225140596)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}\). Moreover, take three events, \(A = \{6, 7, 9, 12\}\), \(B = \{4, 5, 7, 12\}\), and \(C = \{5, 8, 9, 10\}\).
Which of the following elements comprise \(A\cup B\)? (Note, select all that apply).
(Question ID: 0986108733)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\). Moreover, take three events, \(A = \{8, 13, 15, 16\}\), \(B = \{7, 8, 9, 14\}\), and \(C = \{7, 11, 15, 16\}\).
Which of the following elements comprise \(A^C\cup B\)? (Note, select all that apply).
(Question ID: 0100980446)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). Moreover, take three events, \(A = \{1, 7, 9, 10\}\), \(B = \{2, 4, 5, 10\}\), and \(C = \{3, 4, 7, 9\}\).
Which of the following elements comprise \(A^C\)? (Note, select all that apply).
(Question ID: 0416002922)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\). Moreover, take three events, \(A = \{11, 12, 15, 19\}\), \(B = \{11, 13, 14, 16\}\), and \(C = \{10, 11, 13, 14\}\).
Which of the following elements comprise \(B\cup C^C\)? (Note, select all that apply).
(Question ID: 0512146821)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}\). Moreover, take three events, \(A = \{10, 11, 13, 17\}\), \(B = \{10, 14, 16, 17\}\), and \(C = \{11, 12, 13, 16\}\).
Which of the following elements comprise \(A\cup C\)? (Note, select all that apply).
(Question ID: 0475632894)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}\). Moreover, take three events, \(A = \{2, 5, 9, 10\}\), \(B = \{2, 3, 6, 7\}\), and \(C = \{3, 4, 9, 11\}\).
Which of the following elements comprise \(A\cup C\)? (Note, select all that apply).
(Question ID: 0332929845)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\). Moreover, take three events, \(A = \{9, 11, 13, 14\}\), \(B = \{6, 9, 12, 14\}\), and \(C = \{6, 7, 14, 15\}\).
Which of the following elements comprise \(A\cap B^C\)? (Note, select all that apply).
(Question ID: 0974252927)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}\). Moreover, take three events, \(A = \{2, 4, 5, 8\}\), \(B = \{2, 5, 9, 10\}\), and \(C = \{4, 5, 6, 9\}\).
Which of the following elements comprise \(B^C\)? (Note, select all that apply).
(Question ID: 0737689810)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Moreover, take three events, \(A = \{0, 2, 5, 8\}\), \(B = \{0, 3, 6, 7\}\), and \(C = \{0, 3, 7, 9\}\).
Which of the following elements comprise \(A\cup B^C\)? (Note, select all that apply).
(Question ID: 0468902564)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Moreover, take three events, \(A = \{0, 3, 5, 9\}\), \(B = \{0, 3, 6, 7\}\), and \(C = \{0, 4, 6, 9\}\).
Which of the following elements comprise \(B\cap C^C\)? (Note, select all that apply).
(Question ID: 0923848856)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}\). Moreover, take three events, \(A = \{4, 6, 7, 10\}\), \(B = \{7, 8, 10, 11\}\), and \(C = \{6, 7, 10, 11\}\).
Which of the following elements comprise \(A^C\cup B\)? (Note, select all that apply).
(Question ID: 0632131874)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}\). Moreover, take three events, \(A = \{9, 10, 11, 13\}\), \(B = \{8, 9, 10, 16\}\), and \(C = \{8, 15, 16, 17\}\).
Which of the following elements comprise \(B^C\cup C\)? (Note, select all that apply).
(Question ID: 0707574191)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\). Moreover, take three events, \(A = \{8, 9, 13, 14\}\), \(B = \{10, 11, 12, 13\}\), and \(C = \{10, 11, 13, 14\}\).
Which of the following elements comprise \(A\cap B^C\)? (Note, select all that apply).
(Question ID: 0802392101)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{9, 10, 11, 12, 13, 14, 15, 16, 17, 18\}\). Moreover, take three events, \(A = \{12, 15, 17, 18\}\), \(B = \{12, 13, 14, 17\}\), and \(C = \{9, 12, 15, 16\}\).
Which of the following elements comprise \(A\cup B\)? (Note, select all that apply).
(Question ID: 0336110463)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Moreover, take three events, \(A = \{0, 4, 5, 7\}\), \(B = \{0, 2, 6, 7\}\), and \(C = \{1, 3, 4, 6\}\).
Which of the following elements comprise \(B^C\)? (Note, select all that apply).
(Question ID: 0369886306)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\). Moreover, take three events, \(A = \{6, 7, 10, 14\}\), \(B = \{9, 10, 14, 15\}\), and \(C = \{6, 7, 12, 14\}\).
Which of the following elements comprise \(A^C\cap C\)? (Note, select all that apply).
(Question ID: 0928394913)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{9, 10, 11, 12, 13, 14, 15, 16, 17, 18\}\). Moreover, take three events, \(A = \{9, 10, 11, 13\}\), \(B = \{10, 12, 15, 16\}\), and \(C = \{12, 14, 17, 18\}\).
Which of the following elements comprise \(A\cap C\)? (Note, select all that apply).
(Question ID: 0193642299)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\). Moreover, take three events, \(A = \{15, 16, 17, 18\}\), \(B = \{10, 16, 17, 19\}\), and \(C = \{10, 12, 13, 15\}\).
Which of the following elements comprise \(B\cap C\)? (Note, select all that apply).
(Question ID: 0979331221)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{5, 6, 7, 8, 9, 10, 11, 12, 13, 14\}\). Moreover, take three events, \(A = \{5, 7, 11, 13\}\), \(B = \{5, 7, 8, 13\}\), and \(C = \{6, 7, 11, 12\}\).
Which of the following elements comprise \(B^C\cup C\)? (Note, select all that apply).
(Question ID: 0285842660)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}\). Moreover, take three events, \(A = \{8, 12, 14, 17\}\), \(B = \{10, 13, 14, 15\}\), and \(C = \{9, 15, 16, 17\}\).
Which of the following elements comprise \(B\cup C^C\)? (Note, select all that apply).
(Question ID: 0940569136)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{5, 6, 7, 8, 9, 10, 11, 12, 13, 14\}\). Moreover, take three events, \(A = \{5, 9, 12, 14\}\), \(B = \{5, 8, 9, 11\}\), and \(C = \{5, 6, 9, 14\}\).
Which of the following elements comprise \(A\cup B\)? (Note, select all that apply).
(Question ID: 0869287015)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). Moreover, take three events, \(A = \{6, 7, 8, 10\}\), \(B = \{1, 4, 6, 10\}\), and \(C = \{2, 3, 5, 6\}\).
Which of the following elements comprise \(B\cap C\)? (Note, select all that apply).
(Question ID: 0520053210)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{9, 10, 11, 12, 13, 14, 15, 16, 17, 18\}\). Moreover, take three events, \(A = \{9, 10, 11, 18\}\), \(B = \{10, 13, 17, 18\}\), and \(C = \{11, 12, 14, 15\}\).
Which of the following elements comprise \(B^C\cup C\)? (Note, select all that apply).
(Question ID: 0274675214)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\). Moreover, take three events, \(A = \{11, 12, 15, 18\}\), \(B = \{11, 12, 14, 16\}\), and \(C = \{10, 11, 14, 16\}\).
Which of the following elements comprise \(A^C\cup C\)? (Note, select all that apply).
(Question ID: 0505959293)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\). Moreover, take three events, \(A = \{10, 13, 15, 18\}\), \(B = \{13, 17, 18, 19\}\), and \(C = \{11, 14, 16, 17\}\).
Which of the following elements comprise \(B\cap C^C\)? (Note, select all that apply).
(Question ID: 0589273847)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{9, 10, 11, 12, 13, 14, 15, 16, 17, 18\}\). Moreover, take three events, \(A = \{9, 13, 16, 18\}\), \(B = \{9, 12, 14, 15\}\), and \(C = \{10, 11, 12, 16\}\).
Which of the following elements comprise \(A\cup C\)? (Note, select all that apply).
(Question ID: 0549440573)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). Moreover, take three events, \(A = \{4, 6, 7, 9\}\), \(B = \{1, 2, 4, 7\}\), and \(C = \{2, 4, 5, 9\}\).
Which of the following elements comprise \(A^C\cap B\)? (Note, select all that apply).
(Question ID: 0652541556)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}\). Moreover, take three events, \(A = \{8, 10, 11, 16\}\), \(B = \{8, 13, 16, 17\}\), and \(C = \{12, 14, 15, 17\}\).
Which of the following elements comprise \(A^C\cap C\)? (Note, select all that apply).
(Question ID: 0791249784)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\). Moreover, take three events, \(A = \{13, 14, 16, 19\}\), \(B = \{10, 12, 15, 17\}\), and \(C = \{10, 15, 16, 17\}\).
Which of the following elements comprise \(A^C\cup C\)? (Note, select all that apply).
(Question ID: 0073412635)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}\). Moreover, take three events, \(A = \{2, 4, 9, 10\}\), \(B = \{3, 5, 6, 10\}\), and \(C = \{3, 5, 9, 10\}\).
Which of the following elements comprise \(B\cap C\)? (Note, select all that apply).
(Question ID: 0965189537)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Moreover, take three events, \(A = \{2, 5, 6, 8\}\), \(B = \{0, 1, 2, 5\}\), and \(C = \{3, 4, 5, 6\}\).
Which of the following elements comprise \(A\cap B^C\)? (Note, select all that apply).
(Question ID: 0557097666)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}\). Moreover, take three events, \(A = \{3, 4, 6, 11\}\), \(B = \{3, 5, 7, 10\}\), and \(C = \{5, 7, 9, 11\}\).
Which of the following elements comprise \(A^C\)? (Note, select all that apply).
(Question ID: 0049092148)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}\). Moreover, take three events, \(A = \{10, 11, 13, 16\}\), \(B = \{9, 12, 14, 16\}\), and \(C = \{8, 10, 14, 15\}\).
Which of the following elements comprise \(B\cup C^C\)? (Note, select all that apply).
(Question ID: 0158528048)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\). Moreover, take three events, \(A = \{9, 12, 14, 16\}\), \(B = \{8, 14, 15, 16\}\), and \(C = \{7, 12, 15, 16\}\).
Which of the following elements comprise \(A^C\cup C\)? (Note, select all that apply).
(Question ID: 0877550904)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\). Moreover, take three events, \(A = \{7, 8, 9, 10\}\), \(B = \{8, 12, 13, 15\}\), and \(C = \{9, 12, 14, 16\}\).
Which of the following elements comprise \(A^C\cup C\)? (Note, select all that apply).
(Question ID: 0940942135)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{5, 6, 7, 8, 9, 10, 11, 12, 13, 14\}\). Moreover, take three events, \(A = \{6, 8, 12, 13\}\), \(B = \{7, 8, 10, 11\}\), and \(C = \{7, 10, 11, 14\}\).
Which of the following elements comprise \(A^C\cap C\)? (Note, select all that apply).
(Question ID: 0360150071)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\). Moreover, take three events, \(A = \{7, 9, 12, 14\}\), \(B = \{9, 10, 14, 16\}\), and \(C = \{7, 14, 15, 16\}\).
Which of the following elements comprise \(A\cap B\)? (Note, select all that apply).
(Question ID: 0611512885)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\). Moreover, take three events, \(A = \{7, 10, 12, 15\}\), \(B = \{7, 10, 12, 15\}\), and \(C = \{7, 10, 11, 15\}\).
Which of the following elements comprise \(B^C\cap C\)? (Note, select all that apply).
(Question ID: 0201385074)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}\). Moreover, take three events, \(A = \{7, 9, 10, 12\}\), \(B = \{5, 9, 10, 11\}\), and \(C = \{9, 11, 12, 13\}\).
Which of the following elements comprise \(A\cup C\)? (Note, select all that apply).
(Question ID: 0061221952)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\). Moreover, take three events, \(A = \{7, 8, 9, 14\}\), \(B = \{8, 11, 13, 16\}\), and \(C = \{7, 13, 14, 15\}\).
Which of the following elements comprise \(A\cup C\)? (Note, select all that apply).
(Question ID: 0541708378)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\). Moreover, take three events, \(A = \{9, 10, 13, 14\}\), \(B = \{7, 12, 15, 16\}\), and \(C = \{7, 9, 11, 14\}\).
Which of the following elements comprise \(B^C\cap C\)? (Note, select all that apply).
(Question ID: 0736619342)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\). Moreover, take three events, \(A = \{6, 9, 11, 15\}\), \(B = \{9, 10, 12, 15\}\), and \(C = \{6, 11, 12, 13\}\).
Which of the following elements comprise \(A^C\cap B\)? (Note, select all that apply).
(Question ID: 0719808329)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}\). Moreover, take three events, \(A = \{7, 10, 12, 13\}\), \(B = \{4, 6, 9, 11\}\), and \(C = \{4, 7, 8, 9\}\).
Which of the following elements comprise \(A^C\cup B\)? (Note, select all that apply).
(Question ID: 0033746550)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}\). Moreover, take three events, \(A = \{8, 9, 10, 12\}\), \(B = \{5, 10, 11, 12\}\), and \(C = \{4, 5, 6, 9\}\).
Which of the following elements comprise \(A\cup B^C\)? (Note, select all that apply).
(Question ID: 0762385551)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). Moreover, take three events, \(A = \{2, 3, 4, 6\}\), \(B = \{1, 3, 4, 7\}\), and \(C = \{1, 4, 5, 8\}\).
Which of the following elements comprise \(A^C\)? (Note, select all that apply).
(Question ID: 0305122843)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\). Moreover, take three events, \(A = \{6, 7, 8, 10\}\), \(B = \{7, 9, 13, 15\}\), and \(C = \{6, 7, 10, 12\}\).
Which of the following elements comprise \(A^C\)? (Note, select all that apply).
(Question ID: 0856704369)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}\). Moreover, take three events, \(A = \{6, 8, 9, 12\}\), \(B = \{3, 4, 7, 8\}\), and \(C = \{7, 8, 9, 12\}\).
Which of the following elements comprise \(B\cap C\)? (Note, select all that apply).
(Question ID: 0713481441)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{9, 10, 11, 12, 13, 14, 15, 16, 17, 18\}\). Moreover, take three events, \(A = \{10, 14, 16, 17\}\), \(B = \{13, 14, 17, 18\}\), and \(C = \{10, 11, 14, 17\}\).
Which of the following elements comprise \(A^C\cup C\)? (Note, select all that apply).
(Question ID: 0430746826)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{5, 6, 7, 8, 9, 10, 11, 12, 13, 14\}\). Moreover, take three events, \(A = \{8, 10, 13, 14\}\), \(B = \{5, 8, 11, 12\}\), and \(C = \{6, 8, 12, 14\}\).
Which of the following elements comprise \(C^C\)? (Note, select all that apply).
(Question ID: 0486525502)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{9, 10, 11, 12, 13, 14, 15, 16, 17, 18\}\). Moreover, take three events, \(A = \{11, 12, 13, 16\}\), \(B = \{12, 13, 14, 17\}\), and \(C = \{10, 11, 12, 16\}\).
Which of the following elements comprise \(A\cap C\)? (Note, select all that apply).
(Question ID: 0966881842)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}\). Moreover, take three events, \(A = \{2, 6, 7, 10\}\), \(B = \{4, 7, 10, 11\}\), and \(C = \{3, 4, 10, 11\}\).
Which of the following elements comprise \(A^C\)? (Note, select all that apply).
(Question ID: 0392324944)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). Moreover, take three events, \(A = \{1, 2, 4, 8\}\), \(B = \{3, 5, 6, 7\}\), and \(C = \{4, 5, 7, 9\}\).
Which of the following elements comprise \(A\cup C^C\)? (Note, select all that apply).
(Question ID: 0253574991)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\). Moreover, take three events, \(A = \{11, 13, 15, 18\}\), \(B = \{11, 13, 16, 17\}\), and \(C = \{10, 11, 15, 17\}\).
Which of the following elements comprise \(B\cup C\)? (Note, select all that apply).
(Question ID: 0157689443)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}\). Moreover, take three events, \(A = \{4, 6, 9, 13\}\), \(B = \{4, 7, 9, 11\}\), and \(C = \{6, 9, 12, 13\}\).
Which of the following elements comprise \(B\cup C^C\)? (Note, select all that apply).
(Question ID: 0249958676)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). Moreover, take three events, \(A = \{2, 3, 9, 10\}\), \(B = \{1, 4, 8, 10\}\), and \(C = \{1, 3, 5, 8\}\).
Which of the following elements comprise \(A^C\cup C\)? (Note, select all that apply).
(Question ID: 0604625294)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\). Moreover, take three events, \(A = \{6, 8, 10, 14\}\), \(B = \{7, 8, 12, 14\}\), and \(C = \{6, 7, 9, 15\}\).
Which of the following elements comprise \(A\cap B\)? (Note, select all that apply).
(Question ID: 0931331077)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\). Moreover, take three events, \(A = \{8, 9, 10, 13\}\), \(B = \{6, 8, 10, 14\}\), and \(C = \{6, 7, 10, 12\}\).
Which of the following elements comprise \(A^C\cup C\)? (Note, select all that apply).
(Question ID: 0094691184)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Moreover, take three events, \(A = \{2, 4, 6, 9\}\), \(B = \{0, 1, 3, 7\}\), and \(C = \{2, 4, 5, 6\}\).
Which of the following elements comprise \(B\cup C\)? (Note, select all that apply).
(Question ID: 0057258020)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Moreover, take three events, \(A = \{3, 4, 8, 9\}\), \(B = \{0, 3, 7, 9\}\), and \(C = \{1, 2, 3, 6\}\).
Which of the following elements comprise \(A^C\cup C\)? (Note, select all that apply).
(Question ID: 0318817573)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\). Moreover, take three events, \(A = \{6, 8, 9, 11\}\), \(B = \{6, 10, 14, 15\}\), and \(C = \{7, 8, 9, 14\}\).
Which of the following elements comprise \(B^C\cup C\)? (Note, select all that apply).
(Question ID: 0313623908)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\). Moreover, take three events, \(A = \{9, 10, 14, 16\}\), \(B = \{7, 11, 14, 15\}\), and \(C = \{7, 9, 11, 14\}\).
Which of the following elements comprise \(B^C\)? (Note, select all that apply).
(Question ID: 0004728038)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}\). Moreover, take three events, \(A = \{5, 7, 8, 9\}\), \(B = \{3, 5, 7, 12\}\), and \(C = \{4, 5, 10, 12\}\).
Which of the following elements comprise \(A\cup C\)? (Note, select all that apply).
(Question ID: 0851619740)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}\). Moreover, take three events, \(A = \{11, 13, 16, 17\}\), \(B = \{10, 13, 16, 17\}\), and \(C = \{8, 13, 15, 17\}\).
Which of the following elements comprise \(B^C\cup C\)? (Note, select all that apply).
(Question ID: 0177384603)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\). Moreover, take three events, \(A = \{8, 10, 12, 13\}\), \(B = \{8, 10, 14, 15\}\), and \(C = \{6, 7, 12, 13\}\).
Which of the following elements comprise \(B^C\)? (Note, select all that apply).
(Question ID: 0132223985)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}\). Moreover, take three events, \(A = \{4, 5, 8, 11\}\), \(B = \{3, 7, 8, 9\}\), and \(C = \{2, 6, 7, 10\}\).
Which of the following elements comprise \(C^C\)? (Note, select all that apply).
(Question ID: 0388385652)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\). Moreover, take three events, \(A = \{7, 9, 10, 12\}\), \(B = \{7, 9, 11, 13\}\), and \(C = \{8, 9, 14, 15\}\).
Which of the following elements comprise \(A\cap C\)? (Note, select all that apply).
(Question ID: 0838611905)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\). Moreover, take three events, \(A = \{14, 15, 16, 19\}\), \(B = \{11, 12, 15, 19\}\), and \(C = \{11, 12, 14, 16\}\).
Which of the following elements comprise \(A\cap C\)? (Note, select all that apply).
(Question ID: 0449937513)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}\). Moreover, take three events, \(A = \{5, 6, 7, 11\}\), \(B = \{4, 5, 6, 13\}\), and \(C = \{6, 9, 10, 12\}\).
Which of the following elements comprise \(A\cup C\)? (Note, select all that apply).
(Question ID: 0952218046)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\). Moreover, take three events, \(A = \{11, 13, 16, 19\}\), \(B = \{10, 14, 16, 17\}\), and \(C = \{14, 16, 17, 19\}\).
Which of the following elements comprise \(A\cap C^C\)? (Note, select all that apply).
(Question ID: 0743262861)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\). Moreover, take three events, \(A = \{10, 16, 17, 19\}\), \(B = \{11, 12, 13, 15\}\), and \(C = \{12, 13, 17, 19\}\).
Which of the following elements comprise \(A\cup C\)? (Note, select all that apply).
(Question ID: 0639534937)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\). Moreover, take three events, \(A = \{8, 9, 10, 13\}\), \(B = \{9, 12, 13, 15\}\), and \(C = \{6, 8, 10, 12\}\).
Which of the following elements comprise \(A\cup B\)? (Note, select all that apply).
(Question ID: 0418916700)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{9, 10, 11, 12, 13, 14, 15, 16, 17, 18\}\). Moreover, take three events, \(A = \{12, 13, 16, 17\}\), \(B = \{10, 13, 14, 17\}\), and \(C = \{12, 14, 16, 17\}\).
Which of the following elements comprise \(B\cap C\)? (Note, select all that apply).
(Question ID: 0788268133)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\). Moreover, take three events, \(A = \{6, 7, 11, 15\}\), \(B = \{7, 11, 13, 14\}\), and \(C = \{6, 7, 8, 13\}\).
Which of the following elements comprise \(B\cup C^C\)? (Note, select all that apply).
(Question ID: 0778838919)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\). Moreover, take three events, \(A = \{11, 14, 15, 17\}\), \(B = \{12, 14, 15, 17\}\), and \(C = \{12, 13, 14, 16\}\).
Which of the following elements comprise \(A^C\cap B\)? (Note, select all that apply).
(Question ID: 0323233330)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Moreover, take three events, \(A = \{1, 3, 6, 9\}\), \(B = \{1, 2, 6, 7\}\), and \(C = \{1, 3, 7, 9\}\).
Which of the following elements comprise \(B\cup C\)? (Note, select all that apply).
(Question ID: 0139605023)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\). Moreover, take three events, \(A = \{6, 9, 12, 15\}\), \(B = \{6, 7, 9, 15\}\), and \(C = \{7, 8, 10, 11\}\).
Which of the following elements comprise \(B^C\cup C\)? (Note, select all that apply).
(Question ID: 0290714188)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}\). Moreover, take three events, \(A = \{4, 6, 10, 11\}\), \(B = \{5, 9, 11, 12\}\), and \(C = \{5, 9, 10, 13\}\).
Which of the following elements comprise \(B\cup C\)? (Note, select all that apply).
(Question ID: 0387897147)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{9, 10, 11, 12, 13, 14, 15, 16, 17, 18\}\). Moreover, take three events, \(A = \{9, 10, 11, 12\}\), \(B = \{12, 14, 15, 18\}\), and \(C = \{10, 12, 17, 18\}\).
Which of the following elements comprise \(A\cup B\)? (Note, select all that apply).
(Question ID: 0836300348)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}\). Moreover, take three events, \(A = \{9, 11, 14, 16\}\), \(B = \{8, 10, 13, 17\}\), and \(C = \{10, 11, 14, 16\}\).
Which of the following elements comprise \(A\cap C\)? (Note, select all that apply).
(Question ID: 0373517324)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\). Moreover, take three events, \(A = \{8, 10, 11, 15\}\), \(B = \{11, 14, 15, 16\}\), and \(C = \{8, 9, 10, 12\}\).
Which of the following elements comprise \(A\cap B^C\)? (Note, select all that apply).
(Question ID: 0512739951)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). Moreover, take three events, \(A = \{1, 6, 8, 9\}\), \(B = \{3, 5, 7, 9\}\), and \(C = \{1, 5, 7, 10\}\).
Which of the following elements comprise \(A\cup C\)? (Note, select all that apply).
(Question ID: 0894460208)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Moreover, take three events, \(A = \{1, 6, 8, 9\}\), \(B = \{0, 1, 3, 4\}\), and \(C = \{6, 7, 8, 9\}\).
Which of the following elements comprise \(B\cup C\)? (Note, select all that apply).
(Question ID: 0459980088)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}\). Moreover, take three events, \(A = \{4, 6, 9, 12\}\), \(B = \{4, 9, 10, 11\}\), and \(C = \{5, 6, 11, 12\}\).
Which of the following elements comprise \(B\cap C^C\)? (Note, select all that apply).
(Question ID: 0398541287)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). Moreover, take three events, \(A = \{1, 4, 8, 10\}\), \(B = \{3, 4, 5, 7\}\), and \(C = \{6, 8, 9, 10\}\).
Which of the following elements comprise \(A\cap C\)? (Note, select all that apply).
(Question ID: 0646168782)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}\). Moreover, take three events, \(A = \{4, 7, 9, 10\}\), \(B = \{4, 8, 11, 13\}\), and \(C = \{8, 10, 11, 12\}\).
Which of the following elements comprise \(A\cap B^C\)? (Note, select all that apply).
(Question ID: 0119915843)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Moreover, take three events, \(A = \{3, 5, 7, 9\}\), \(B = \{1, 6, 7, 9\}\), and \(C = \{0, 5, 6, 7\}\).
Which of the following elements comprise \(A\cup B\)? (Note, select all that apply).
(Question ID: 0116966707)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\). Moreover, take three events, \(A = \{7, 10, 12, 15\}\), \(B = \{10, 12, 15, 16\}\), and \(C = \{12, 13, 14, 15\}\).
Which of the following elements comprise \(A\cup C\)? (Note, select all that apply).
(Question ID: 0890333563)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). Moreover, take three events, \(A = \{1, 3, 4, 9\}\), \(B = \{1, 4, 7, 9\}\), and \(C = \{1, 2, 3, 5\}\).
Which of the following elements comprise \(A^C\)? (Note, select all that apply).
(Question ID: 0619693690)
Suppose that the sample space for an experiment, \(\mathcal{S}\) is given by \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Moreover, take three events, \(A = \{2, 3, 5, 6\}\), \(B = \{1, 5, 7, 9\}\), and \(C = \{3, 4, 5, 9\}\).
Which of the following elements comprise \(A\cup B^C\)? (Note, select all that apply).
(Question ID: 0388831385)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): at least one screen is not correctly calibrated; and \(B\): screens \(5\) and \(8\) are correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0748383088)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): at least one screen is not correctly calibrated; and \(B\): exactly one screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0963338615)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): all screens are correctly calibrated; and \(B\): the third screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0137960460)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): at least one screen is not correctly calibrated; and \(B\): screens \(5\) and \(8\) are correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0232773340)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): at least one screen is not correctly calibrated; and \(B\): exactly one screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0597735131)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): all screens are correctly calibrated; and \(B\): at least one screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0731751742)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): all screens are correctly calibrated; and \(B\): exactly one screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0990912357)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): at least one screen is not correctly calibrated; and \(B\): exactly one screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0062778682)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): the third screen is not correctly calibrated; and \(B\): exactly one screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0858574946)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): at least one screen is not correctly calibrated; and \(B\): exactly one screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0174879612)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): only the even numbered screens are correctly calibrated; and \(B\): screens \(5\) and \(8\) are correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0086531192)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): the third screen is not correctly calibrated; and \(B\): at least one screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0463856513)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): only the even numbered screens are correctly calibrated; and \(B\): the third screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0948086400)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): the third screen is not correctly calibrated; and \(B\): screens \(5\) and \(8\) are correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0342205469)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): all screens are correctly calibrated; and \(B\): exactly one screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0679937316)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): all screens are correctly calibrated; and \(B\): the third screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0671063110)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): at least one screen is not correctly calibrated; and \(B\): screens \(5\) and \(8\) are correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0091681861)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): only the even numbered screens are correctly calibrated; and \(B\): the third screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0536185810)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): at least one screen is not correctly calibrated; and \(B\): screens \(5\) and \(8\) are not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0761326411)
A cinema has \(12\) screens, numbered \(1\) through \(12\). Before opening, an employee checks to ensure that the projectors are correctly calibrated.
Consider the following events:
\(A\): the third screen is not correctly calibrated; and \(B\): at least one screen is not correctly calibrated.
Are these events disjoint, or not disjoint?
(Question ID: 0973829064)
Note that the symbol \(\subseteq\) is the subset or equal symbol. If we write \(A \subseteq B\), then \(A\) is either a subset of \(B\) or else \(A\) is equal to \(B\). This is in contrast to using \(A \subseteq B\) which suggests that \(A\) is not equal to \(B\). The distinction is analogous to the difference between writing \(x < y\) versus \(x \leq y\). Throughout these notes we will predominantly use \(\subseteq\).↩︎
When speaking to a statistician, they would understand “Charles has to pay” as an event that can occur based on the defined sample space, by simply transforming it into the language of the sample space. However, the distinction is important to make: events are always subsets of the sample space. Once this is second nature, it is a rule that can be loosened, as the knowledge can always be fallen back on when needed. Simply put: you need to know the rules in order to break them!↩︎
While both or and not language is likely clear from the examples we have seen so far, and language may be slightly less obvious. While we will explore this in more depth shortly, note that you could not have two simple events occurring simultaneously. If \(E_1\) and \(E_2\) are both simple events, then you can have \(E_1\) or \(E_2\), and you can have not \(E_1\), but you cannot have \(E_1\) and \(E_2\). This is not true for compound events.↩︎
It may be helpful to notice that you can mix and match these terms to your hearts content!↩︎
Note that whenever we have two events, \(A\) and \(B\), with \(A\subseteq B\), then \(A\cup B = B\).↩︎
Note that whenever we have two events, \(A\) and \(B\), with \(A\subseteq B\) then \(A \cap B = A\).↩︎
Note, these three are compound events with \(13\) different outcomes contained within them.↩︎
These are all compound events with four different options.↩︎
R does not implement complements directly, and instead implements the set difference operation. The set difference function,
setdiff(x, y)returns the set of all elements inxwhich are not iny, a sort of subtracting of sets. The complement of a set is defined to be \(A^C = \text{setdiff}(\mathcal{S}, A)\), indicating why this works!↩︎This is a general principle in mathematics. Coming up with one example that makes something seem true does not form an argument demonstrating that it is true. Venn Diagrams should largely be thought of as specific examples of the underlying phenomena, which are great if you’re a visual learner!↩︎