Answer to Question #205141 in Statistics and Probability for Shema Derrick

Question

. For married couples living in a certain suburb, the probability that the husband will vote on a bond referendum is 0.21, the probability that his wife will vote in the referendum is 0.28, and the probability that both the husband and wife will vote is 0.15. What is the probability that: a) at least one member of a married couple will vote? b) a wife will vote, given that her husband will vote? c) a husband will vote, given that his wife does not vote?

Answer 1

Let $H$ denote the event that the husband will vote on a bond referendum.

Let $W$ denote the event that his wife will vote on a bond referendum.

Given $P(H)=0.21, P(W)=0.28, P(H\cap W)=0.15.$

a)

$$P(H\cup W)=P(H)+P(W)-P(H\cap W)$$

$$=0.21+0.28-0.15=0.24$$

The probability that at least one member of a married couple will vote is 0.24.

b)

$$P(W|H)=\dfrac{P(H\cap W)}{P(H)}=\dfrac{0.15}{0.21}=\dfrac{5}{7}$$

The probability that a wife will vote, given that her husband will vote is $\dfrac{5}{7}.$

c)

$$P(W')=1-P(W)=1-0.28=0.72$$

$$P(H\cap W')=P(H)-P(H\cap W)$$

$$=0.21-0.15=0.06$$

$$P(H|W')=\dfrac{P(H\cap W')}{P(W')}=\dfrac{0.06}{0.72}=\dfrac{1}{12}$$

The probability that a husband will vote, given that his wife does not vote is $\dfrac{1}{12} .$