91Ó°ÊÓ

In a past presidential election, it was estimated that the probability that the Republican candidate would be elected was \(\frac{3}{5}\), and therefore the probability that the Democratic candidate would be elected was \(\frac{2}{5}\) (the two Independent candidates were given no chance of being elected). It was also estimated that if the Republican candidate were elected, the probability that a conservative, moderate, or liberal judge would be appointed to the Supreme Court (one retirement was expected during the presidential term) was \(\frac{1}{2}, \frac{1}{3}\), and \(\frac{1}{6}\), respectively. If the Democratic candidate were elected, the probabilities that a conservative, moderate, or liberal judge would be appointed to the Supreme Court would be \(\frac{1}{8}, \frac{3}{8}\), and \(\frac{1}{2}\), respectively. A conservative judge was appointed to the Supreme Court during the presidential term. What is the probability that the Democratic candidate was elected?

Step by step solution

01

Identify the Known Probabilities

There are several probabilities provided in the problem. The probability that the Republican candidate would be elected is \(\frac{3}{5}\), and the probability that the Democratic candidate would be elected is \(\frac{2}{5}\). Lastly, the probabilities of appointing different types of judges are given for both candidates. We are specifically interested in the probabilities associated with appointing a conservative judge.

02

Identify the Events and their Probabilities

Define the following events. Let's call D the event that the Democratic candidate is elected (probability \(P(D)=\frac{2}{5}\)), and C be the event that a conservative judge is appointed. The probability that a conservative judge is appointed given that the Democratic candidate is elected is \(P(C|D)=\frac{1}{8}\).

03

Calculate the Total Probability of Appointing a Conservative Judge

The total probability of appointing a conservative judge would be the sum of the probabilities of appointing a conservative judge given each candidate is elected, weighted by the probability of each candidate being elected. This would be: \[P(C) = P(D) \times P(C|D) + P(R) \times P(C|R)\] where R is the event that the Republican candidate is elected (\(P(R)=\frac{3}{5}\)), and \(P(C|R)=\frac{1}{2}\). Substituting the given values, we get: \[P(C) = \frac{2}{5} \times \frac{1}{8} + \frac{3}{5} \times \frac{1}{2} = \frac{7}{20}\]

04

Use Bayes' Theorem to Find the Desired Probability

Finally, use Bayes' Theorem to find the probability that the Democratic candidate is elected given that a conservative judge is appointed. Bayes' Theorem is: \[P(D|C) = \frac{P(D) \times P(C|D)}{P(C)}\] Substituting the values we have found, we get: \[P(D|C) = \frac{\frac{2}{5} \times \frac{1}{8}}{\frac{7}{20}} = \frac{1}{14}\] Therefore, the probability that the Democratic candidate was elected given that a conservative judge was appointed to the Supreme Court is \(\frac{1}{14}\).

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