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Chapter 7: Problem 54

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) and \(B\) are mutually exclusive and \(P(B) \neq 0\), then \(P(A \mid B)=0 .\)

Short Answer

Expert verified
The statement is true. If \(A\) and \(B\) are mutually exclusive and \(P(B) \neq 0\), then \(P(A \mid B) = 0\). This is because mutually exclusive events cannot occur simultaneously, making the conditional probability of one event given the occurrence of the other event equal to zero.

Step by step solution

01

Recall the formula for conditional probability and mutually exclusive events

The conditional probability of event \(A\) given event \(B\) is denoted as \(P(A \mid B)\) and is calculated using the formula: \[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\] Additionally, for two mutually exclusive events (events that cannot occur simultaneously), the probability of their intersection \(P(A \cap B)\) is equal to \(0\).
02

Applying the formula for conditional probability to the given statement

Since \(A\) and \(B\) are mutually exclusive, we have: \[P(A \cap B) = 0\] Now we substitute this into the conditional probability formula for \(P(A \mid B)\): \[P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{0}{P(B)}\]
03

Evaluate the expression

We are given that \(P(B) \neq 0\). Thus, the denominator in the above expression is non-zero. Evaluating the expression, we get: \[P(A \mid B) = \frac{0}{P(B)} = 0\]
04

Conclusion

The statement is true. If \(A\) and \(B\) are mutually exclusive and \(P(B) \neq 0\), then \(P(A \mid B) = 0\). This is because mutually exclusive events cannot occur simultaneously, making the conditional probability of one event given the occurrence of the other event equal to zero.

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Key Concepts

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

Mutually Exclusive Events
In probability theory, events are considered mutually exclusive when they cannot occur simultaneously. This means that if one event happens, the other cannot. For example, when flipping a coin, the events "landing on heads" and "landing on tails" are mutually exclusive.
When two events are mutually exclusive, the probability of both happening at the same time is zero. Mathematically, this can be expressed as:
  • If events \(A\) and \(B\) are mutually exclusive, then \(P(A \cap B) = 0\).
Understanding this concept is vital because it simplifies the calculation of probabilities in scenarios where such events are identified. Essentially, the intersection of mutually exclusive events yields a probability of zero because their occurrence at the same time is impossible.
Probability Theory
Probability theory is the branch of mathematics that deals with the study of random events and the likelihood of their occurrence. At its core, probability theory provides the tools to quantify and model uncertainty.
Core components of probability theory include:
  • The sample space \(S\): the set of all possible outcomes of an experiment.
  • An event \(A\): a subset of the sample space, which can be a single outcome or a group of outcomes.
  • The probability of an event \(P(A)\), which is a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
Probability theory is fundamental in understanding situations under uncertainty and is used extensively in various fields such as finance, science, and engineering to make informed decisions based on the likelihood of various outcomes.
Event Intersection
The intersection of events involves finding the probability that multiple events occur at the same time. When discussing probability, the intersection is denoted as \(A \cap B\), which represents that both events \(A\) and \(B\) occur simultaneously.
To determine the probability of the intersection of two events, you use the formula for joint probability:
  • For events \(A\) and \(B\), the joint probability is \(P(A \cap B)\).
In the case of mutually exclusive events, this intersection probability is zero because they cannot occur together. Understanding event intersection aids in calculating probabilities when events overlap in time or occurrence.

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Most popular questions from this chapter

Let \(E\) and \(F\) be mutually exclusive events and suppose \(P(F) \neq 0\). Find \(P(E \mid F)\) and interpret your result.

In an online survey of 1962 executives from 64 countries conducted by Korn/Ferry International between August and October 2006, the executives were asked if they would try to influence their children's career choices. Their replies: A (to a very great extent), B (to a great extent), \(\mathrm{C}\) (to some extent), D (to a small extent), and \(\mathrm{E}\) (not at all) are recorded below: $$\begin{array}{lccccc} \hline \text { Answer } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Respondents } & 135 & 404 & 1057 & 211 & 155 \\ \hline \end{array}$$ What is the probability that a randomly selected respondent's answer was \(\mathrm{D}\) (to a small extent) or \(\mathrm{E}\) (not at all)?

A medical test has been designed to detect the presence of a certain disease. Among those who have the disease, the probability that the disease will be detected by the test is \(.95\). However, the probability that the test will erroneously indicate the presence of the disease in those who do not actually have it is .04. It is estimated that \(4 \%\) of the population who take this test have the disease. a. If the test administered to an individual is positive, what is the probability that the person actually has the disease? b. If an individual takes the test twice and both times the test is positive, what is the probability that the person actually has the disease? (Assume that the tests are independent.)

The Office of Admissions and Records of a large western university released the accompanying information concerning the contemplated majors of its freshman class:3 $$\begin{array}{lccc} \text { Major } & \text {This Major, \% } & \text { Females, \% } & \text { Males, \% } \\ \hline \text { Business } & 24 & 38 & 62 \\ \hline \text { Humanities } & 8 & 60 & 40 \\ \hline \text { Education } & 8 & 66 & 34 \\ \hline \text { Social science } & 7 & 58 & 42 \\ \hline \text { Natural sciences } & 9 & 52 & 48 \\ \hline \text { Other } & 44 & 48 & 52 \\ \hline \end{array}$$ What is the probability that a. A student selected at random from the freshman class is a female? b. A business student selected at random from the fresh- man class is a male? c. A female student selected at random from the freshman class is majoring in business?

In a home theater system, the probability that the video components need repair within I \(\mathrm{yr}\) is \(.01\), the probability that the electronic components need repair within I yr is \(.005\), and the probability that the audio components need repair within I yr is \(.001\). Assuming that the events are independent, find the probability that a. At least one of these components will need repair within 1 yr. b. Exactly one of these components will need repair within 1 vr.
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