In a past presidential election, it was estimated that the probability that the Republican candidate would be elected was 2/3, and therefore the probability that the Democratic candidate would be elected was 1/3 (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 1/3, 1/3, and 1/3, 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 1/8, 1/4, and 5/8, respectively. A conservative judge was appointed to the Supreme Court during the presidential term. What is the probability that the Democratic candidate was elected? (Round your answer to three decimal places.)

Question

In a past presidential election, it was estimated that the probability that the Republican candidate would be elected was 2/3, and therefore the probability that the Democratic candidate would be elected was 1/3 (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 1/3, 1/3, and 1/3, 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 1/8, 1/4, and 5/8, respectively. A conservative judge was appointed to the Supreme Court during the presidential term. What is the probability that the Democratic candidate was elected? (Round your answer to three decimal places.)

Answer 1

To find the probability that the Democratic candidate was elected given that a conservative judge was appointed to the Supreme Court, we can use Bayes' theorem.

Let's start by defining the events:
A: The Republican candidate was elected.
D: The Democratic candidate was elected.
C: A conservative judge was appointed to the Supreme Court.

We need to find P(D|C), the probability that the Democratic candidate was elected given that a conservative judge was appointed.

According to Bayes' theorem:
P(D|C) = (P(C|D) * P(D)) / P(C)

Where:
P(D) = Probability that the Democratic candidate was elected = 1/3 (given in the question).
P(C|D) = Probability that a conservative judge was appointed given that the Democratic candidate was elected = 1/8 (given in the question).
P(C) = Probability that a conservative judge was appointed.

To find P(C), we can use the Law of Total Probability:
P(C) = P(C|A) * P(A) + P(C|D) * P(D)

Where:
P(A) = Probability that the Republican candidate was elected = 2/3 (given in the question).
P(C|A) = Probability that a conservative judge was appointed given that the Republican candidate was elected = 1/3 (given in the question).

Substituting the values:
P(C) = (1/3) * (2/3) + (1/8) * (1/3) = 2/9 + 1/24 = 17/72

Now, we can calculate P(D|C):
P(D|C) = (P(C|D) * P(D)) / P(C) = (1/8 * 1/3) / (17/72) = (1/24) / (17/72) = 6/17 ≈ 0.353

Therefore, the probability that the Democratic candidate was elected, given that a conservative judge was appointed to the Supreme Court, is approximately 0.353 (rounded to three decimal places).