According to exit polls from the 2000 presidential election, the probability of a voter identifying as gay (including lesbians) was P (G) = 0.04. The probability of voting for Bush, given that a voter was gay, was P (B given G) = 0.25, and the probability of voting for Bush, given that a voter was not gay, was P (B given not G) = 0.50.

Find P (G intersection B)

Find P (G' intersection B) Find P (B), the overall probability of voting for Bush, keeping in mind that a voter was either gay and voted for Bush or not gay and voted for Bush.

P (G∩B) = 0.01

P (G'∩B) = 0.48

P (B) = 0.49

Step-by-step explanation:

The given probabilities are

P (G) = 0.04

P (B/G) = 0.25

P (B/G') = 0.5

P (G intersection B) = P (G∩B) = ?

According to definition of conditional probability

P (B/G) = P (G∩B) / P (G)

So,

P (G∩B) = P (G) * P (B/G)

P (G∩B) = 0.04*0.25

P (G∩B) = 0.01

Thus, P (G intersection B) = P (G∩B) = 0.01.

Now, P (G' intersection B) = P (G'∩B) = ?

According to definition of conditional probability

P (B/G') = P (G'∩B) / P (G')

So,

P (G'∩B) = P (G') * P (B/G')

P (G') = 1-P (G) = 1-0.04=0.96

P (G'∩B) = 0.96*0.5

P (G'∩B) = 0.48

Thus, P (G' intersection B) = P (G'∩B) = 0.48.

Now, P (B) = ?

We know a voter is either gay and voted for Bush or not gay and voted for Bush so,

P (B) = P (G∩B) + P (G'∩B)

P (B) = 0.01+0.48

P (B) = 0.49

Thus, the overall probability of voting for Bush P (B) = 0.49.