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Chapter 7: Problem 19

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?

Short Answer

Expert verified
The probability that a senator selected at random with military service is a Republican is approximately 42.4%. \(P(Republican | Military Service) 鈮� 0.424\)

Step by step solution

01

Understand the question

We need to find the probability that a randomly chosen senator with military service is Republican. In other words, we need to find the value of the conditional probability P(Republican | Military Service).
02

Calculate the probabilities

First, let's find the probability of selecting a Republican Senator, a Democrat Senator, a Senator who served in the military, and a Senator who did not serve in the military. - Total Number of Senators = 57 Democrats + 43 Republicans = 100 - P(Republican) = Number of Republicans / Total Number of Senators = 43 / 100 - P(Democrat) = Number of Democrats / Total Number of Senators = 57 / 100 - P(Military Service) = (38 Democrats with Military Service + 28 Republicans with Military Service) / Total Number of Senators = (38 + 28) / 100 = 66 / 100 - P(No Military Service) = (57 Democrats - 38 Democrats with Military Service + 43 Republicans - 28 Republicans with Military Service) / Total Number of Senators = (19 + 15) / 100 = 34 / 100
03

Calculate the conditional probabilities

Next, we'll calculate the conditional probabilities that we need for our final calculation. - P(Military Service | Republican) = Number of Republican Senators with Military Service / Total Number of Republicans = 28 / 43 - P(Military Service | Democrat) = Number of Democrat Senators with Military Service / Total Number of Democrats = 38 / 57
04

Use the Bayes' theorem to find the required probability

We can use the Bayes' theorem to find the probability of a senator being a Republican given that they served in the military. Bayes' theorem formula: P(A | B) = (P(B | A) * P(A)) / P(B) In our case, we want to find P(Republican | Military Service). Therefore, A = Republican and B = Military Service. Using the Bayes' theorem, we have: P(Republican | Military Service) = (P(Military Service | Republican) * P(Republican)) / P(Military Service) Plugging in the values, we get: P(Republican | Military Service) = ((28 / 43) * (43 / 100)) / (66 / 100)
05

Calculate the final probability

Now we can calculate the final probability: P(Republican | Military Service) = (28 / 43) * (43 / 66) = 28 / 66 鈮� 0.424 So the probability that a senator selected at random with military service is a Republican is approximately 42.4%.

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