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Chapter 2: Problem 26

Suppose that three events \(-A, B\), and \(C\)-are defined on a sample space \(S\). Use the union, intersection, and complement operations to represent each of the following events: (a) none of the three events occurs (b) all three of the events occur (c) only event \(A\) occurs (d) exactly one event occurs (e) exactly two events occur

Short Answer

Expert verified
The representation of the events are as follows: (a) \(A'\cap B'\cap C'\) (b) \(A\cap B\cap C\) (c) \(A\cap B'\cap C'\) (d) \((A \cap B'\cap C')\cup (A' \cap B\cap C')\cup (A' \cap B'\cap C)\) (e) \((A \cap B\cap C')\cup (A \cap B'\cap C)\cup (A' \cap B\cap C)\)

Step by step solution

01

None of the Events Occur

To represent that none of the events occur, the complementary operation is used for all the three events \(A\), \(B\) and \(C\). Therefore \(A'\cap B'\cap C'\) is used to denote that none of the events occur.
02

All of the Events Occur

To represent that all three events occur, the intersection operation is used for \(A\), \(B\) and \(C\). Therefore, \(A\cap B\cap C\) is used to denote that all of the events occur.
03

Only Event A Occurs

To represent that only event \(A\) occurs, intersection and complement operations are used. Event \(A\) is intersected with the complement of the events \(B\) and \(C\). Therefore, \(A\cap B'\cap C'\) is used to represent that only event \(A\) occurs.
04

Exactly one Event Occurs

To represent this, the union operation is used to show that either of the events occurs. For only one event to occur, it must exclude the occurrence of the other two events. Thus, this is represented as \((A \cap B'\cap C')\cup (A' \cap B\cap C')\cup (A' \cap B'\cap C)\).
05

Exactly Two Events Occur

This is represented as the union of the three possibilities of having exactly two events occurring. Those are: \(A\) and \(B\) occur but not \(C\), \(A\) and \(C\) occur but not \(B\), and \(B\) and \(C\) occur but not \(A\). Thus, this is represented as \((A \cap B\cap C')\cup (A \cap B'\cap C)\cup (A' \cap B\cap C)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complementary Events
In set theory and probability, complementary events are essential concepts for understanding how events relate within a probability space. A complementary event is essentially the opposite of a given event within the sample space. It includes all outcomes that are not part of the original event.For example, if we have an event \(A\), its complement, denoted as \(A'\), includes all outcomes in the sample space \(S\) that do not belong to \(A\). If you know the probability of an event \(A\), you can easily find the probability of the complement of \(A\) by subtracting the probability of \(A\) from 1:\[ P(A') = 1 - P(A) \]Understanding complements is useful for various operations and problem-solving in probability, such as determining the probability that none of several events occur (e.g., \(A' \cap B' \cap C'\)). This scenario describes not experiencing any of the specified events.
Union and Intersection of Events
The concepts of union and intersection are key in dealing with events in probability theory. They are essential for describing occurrences of multiple events.
  • Union: The union of two events \(A\) and \(B\), denoted as \(A \cup B\), includes all outcomes that belong to either \(A\) or \(B\), or both. The union represents the occurrence of at least one of the events.
  • Intersection: The intersection of two events \(A\) and \(B\), denoted as \(A \cap B\), contains all outcomes that are common to both \(A\) and \(B\). The intersection represents the scenarios where both events happen simultaneously.
These operations enable us to express complex probability events such as the occurrence of all specified events (e.g., \(A \cap B \cap C\)) or exactly one event out of a set \((A \cap B' \cap C') \cup (A' \cap B \cap C') \cup (A' \cap B' \cap C)\). Combining unions and intersections helps represent multiple scenarios for event occurrences.
Probability Operations
Probability operations involve the methods used to compute the likelihood of various event combinations. Given a sample space \(S\) and defined events, operations such as complement, union, and intersection help evaluate the probability of these events.
  • Complement: To find the probability of a complementary event, use the formula for the complement, which is based on having all outcomes outside the original event.
  • Union and Intersection: For unions, you must account for any potential overlap by adding the probabilities of individual events and subtracting the probability of their intersection: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Intersection probabilities require the occurrence of all involved events. If the events are independent, the probability of their intersection is the product of their probabilities: \[ P(A \cap B) = P(A) \times P(B) \]
Using these operations allows students to understand the weight or likelihood of complicated scenarios, such as determining probabilities when certain events must or must not happen. Learning these operations equips students with the tools needed to analyze the probability of various outcomes.

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Most popular questions from this chapter

What is the smallest number of switches wired in parallel that will give a probability of at least \(0.98\) that a circuit will be completed? Assume that each switch operates independently and will function properly \(60 \%\) of the time.

Prove that successive terms in the sequence \(\left(\begin{array}{c}n \\\ 0\end{array}\right)\), \(\left(\begin{array}{l}n \\ 1\end{array}\right), \ldots,\left(\begin{array}{l}n \\ n\end{array}\right)\) first increase and then decrease. [Hint: Examine the ratio of two successive terms, \(\left(\begin{array}{c}n \\ j+1\end{array}\right) /\left(\begin{array}{l}n \\\ j\end{array}\right)\).]

Suppose that \(P(A \cap B)=0.2, P(A)=0.6\), and \(P(B)=0.5\). (a) Are \(A\) and \(B\) mutually exclusive? (b) Are \(A\) and \(B\) independent? (c) Find \(P\left(A^{C} \cup B^{C}\right)\).

A man has \(n\) keys on a key ring, one of which opens the door to his apartment. Having celebrated a bit too much one evening, he returns home only to find himself unable to distinguish one key from another. Resourceful, he works out a fiendishly clever plan: He will choose a key at random and try it. If it fails to open the door, he will discard it and choose at random one of the remaining \(n-1\) keys, and so on. Clearly, the probability that he gains entrance with the first key he selects is \(1 / n\). Show that the probability the door opens with the third key he tries is also \(1 / n\). (Hint: What has to happen before he even gets to the third key?)

Brett and Margo have each thought about murdering their rich Uncle Basil in hopes of claiming their inheritance a bit early. Hoping to take advantage of Basil's predilection for immoderate desserts, Brett has put rat poison into the cherries flamb茅; Margo, unaware of Brett's activities, has laced the chocolate mousse with cyanide. Given the amounts likely to be eaten, the probability of the rat poison being fatal is \(0.60 ;\) the cyanide, \(0.90\). Based on other dinners where Basil was presented with the same dessert options, we can assume that he has a \(50 \%\) chance of asking for the cherries flamb茅, a \(40 \%\) chance of ordering the chocolate mousse, and a \(10 \%\) chance of skipping dessert altogether. No sooner are the dishes cleared away than Basil drops dead. In the absence of any other evidence, who should be considered the prime suspect?
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