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Chapter 7: Problem 51

Let \(E\) and \(F\) be events such that \(F \subset E\). Find \(P(E \mid F)\) and interpret your result.

Short Answer

Expert verified
The conditional probability \(P(E \mid F)\) is 1. This means if event F occurs, then event E is certain to occur, since F is a subset of E and all outcomes in F are included in E.

Step by step solution

01

Identify the intersection of events E and F

Since F is a subset of E, it means that all outcomes in event F are also included in event E. Therefore, the intersection of E and F, denoted by E 鈭� F, is simply F itself: \[E \cap F = F\]
02

Calculate the probabilities P(E 鈭� F) and P(F)

Given that \(E \cap F = F\), for the numerator, the probability of the intersection is the same as the probability of F: \[P(E \cap F) = P(F)\] As for the denominator, we don't have any information about P(F) from the given exercise. So, we'll leave it as is, denoting it as P(F).
03

Apply the formula for conditional probability

Now that we have the probabilities for \(P(E \cap F)\) and \(P(F)\), we can apply the formula for conditional probability: \[P(E \mid F) = \frac{P(E \cap F)}{P(F)}\] By substituting \(P(E \cap F) = P(F)\): \[P(E \mid F) = \frac{P(F)}{P(F)}\] Since P(F) > 0, we can simplify this expression: \[P(E \mid F) = 1\]
04

Interpret the result

The result P(E | F) = 1 indicates that if event F occurs, then event E is certain to occur as well. This is consistent with the given condition that F is a subset of E, meaning once we know F has occurred, E must also have occurred since all outcomes in F are included in E.

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Most popular questions from this chapter

In a survey of 1000 eligible voters selected at random, it was found that 80 had a college degree. Additionally, it was found that \(80 \%\) of those who had a college degree voted in the last presidential election, whereas \(55 \%\) of the people who did not have a college degree voted in the last presidential election. Assuming that the poll is representative of all eligible voters, find the probability that an eligible voter selected at random a. Had a college degree and voted in the last presidential election. b. Did not have a college degree and did not vote in the last presidential election. c. Voted in the last presidential election. d. Did not vote in the last presidential election.

The following table, compiled in 2004, gives the percentage of music downloaded from the United States and other countries by U.S. users: $$ \begin{array}{lcccccccc} \hline \text { Country } & \text { U.S. } & \text { Germany } & \text { Canada } & \text { Italy } & \text { U.K. } & \text { France } & \text { Japan } & \text { Other } \\ \hline \text { Percent } & 45.1 & 16.5 & 6.9 & 6.1 & 4.2 & 3.8 & 2.5 & 14.9 \\\ \hline \end{array} $$ a. Verify that the table does give a probability distribution for the experiment. b. What is the probability that a user who downloads music, selected at random, obtained it from either the United States or Canada? c. What is the probability that a U.S. user who downloads music, selected at random, does not obtain it from Italy, the United Kingdom (U.K.), or France?

Asurvey involving 400 likely Democratic voters and 300 likely Republican voters asked the question: Do you support or oppose legislation that would require registration of all handguns? The following results were obtained: $$\begin{array}{lcc} \hline \text { Answer } & \text { Democrats, \% } & \text { Republicans, \% } \\\ \hline \text { Support } & 77 & 59 \\ \hline \text { Oppose } & 14 & 31 \\ \hline \text { Don't know/refused } & 9 & 10 \\ \hline \end{array}$$ If a randomly chosen respondent in the survey answered "oppose," what is the probability that he or she is a likely Democratic voter?

The 1992 U.S. Senate was composed of 57 Democrats and 43 Republicans. Of the Democrats, 38 served in the military, whereas 28 of the Republicans had seen military service. If a senator selected at random had served in the military, what is the probability that he or she was Republican?

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 \(A\) is a subset of \(B\), then \(P(A) \leq P(B)\).
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