In a past presidential election, it was estimated that the probability that the Republican candidate would be elected was 1/3, and therefore the probability that the Democratic candidate would be elected was 2/3 (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 1/6, 1/6, and 2/3, 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 2/9, 5/9, and 2/9, respectively. A conservative judge was appointed to the Supreme Court during the presidential term. What is the probability that the Democratic candidate was elected?

Question

In a past presidential election, it was estimated that the probability that the Republican candidate would be elected was 1/3, and therefore the probability that the Democratic candidate would be elected was 2/3 (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 1/6, 1/6, and 2/3, 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 2/9, 5/9, and 2/9, respectively. A conservative judge was appointed to the Supreme Court during the presidential term. What is the probability that the Democratic candidate was elected?

Answer 1

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

Let's define the events:
A: Democratic candidate was elected.
B: Republican candidate was elected.
C: A conservative judge was appointed to the Supreme Court.

We need to find P(A|C), the probability of event A happening given that C has occurred.

According to Bayes' theorem, we have:

P(A|C) = P(C|A) * P(A) / P(C)

We already know P(C|A) = 2/9, the probability of a conservative judge being appointed given that the Democratic candidate was elected.

We are given that the probability of the Republican candidate being elected is 1/3, so P(B) = 1/3.

To find P(C), the probability of a conservative judge being appointed, we can use the Law of Total Probability:

P(C) = P(C|A) * P(A) + P(C|B) * P(B)

From the given information, P(C|A) = 2/9, P(C|B) = 1/6, and P(B) = 1/3. We need to find P(A).

Given that the probabilities of the Republican and the Democratic candidate being elected are 1/3 and 2/3 respectively, we have P(A) = 2/3.

Now, let's substitute these values into the equation for P(C):

P(C) = (2/9) * (2/3) + (1/6) * (1/3)
= 4/27 + 1/18
= 7/54

Now we can substitute all the values into Bayes' theorem:

P(A|C) = (2/9) * (2/3) / (7/54)
= (4/27) / (7/54)
= (4/27) * (54/7)
≈ 0.2857

Therefore, the probability that the Democratic candidate was elected given that a conservative judge was appointed to the Supreme Court is approximately 0.2857 or 28.57%.