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 the wife will vote on the referendum is 0.28, and the probability that both the husband and the wife will vote is 0.15. What is the probability that

Answer 1

Answer:

a) The probability that  at least one member of a married couple will vote is 0.34.

b) The probability that  a wife will vote, given that her husband will vote is 0.7143.

c) The probability that  a husband will vote, given that his wife will not  vote is 0.0833.

Step-by-step explanation:

Given : 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 the wife will vote on the referendum is 0.28, and the probability that both the husband and the wife will vote is 0.15.

To find : 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 will not  vote?

Solution :

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

i.e. P(H)=0.21

Let W be the event that the wife will vote on a bond referendum.

i.e. P(W)=0.28

The probability that both the husband and the wife will vote is 0.15.

i.e. $P(H\cap W)=0.15$

(a) At least one member of a married couple will vote i.e. $P(H\cup W)$

Applying union of events,

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

$P(H\cup W)=0.21+0.28-0.15$

$P(H\cup W)=0.34$

The probability that  at least one member of a married couple will vote is 0.34.

(b) A wife will vote, given that her husband will vote i.e. $P(W/H)$

Applying to definition of conditional probability,

$P(W/H)=\frac{P(W\cap H)}{P(H)}$

$P(W/H)=\frac{0.15}{0.21}$

$P(W/H)=0.7143$

The probability that  a wife will vote, given that her husband will vote is 0.7143.

(c) A husband will vote, given that his wife will not  vote i.e. $P(H/W')$

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

Applying to definition of conditional probability,

$P(H/W')=\frac{P(H\cap W')}{P(W')}$

$P(H/W')=\frac{P(H)-P(H\cap W)}{P(W')}$

$P(H/W')=\frac{0.21-0.15}{0.72}$

$P(H/W')=\frac{0.06}{0.72}$

$P(H/W')=0.0833$

The probability that  a husband will vote, given that his wife will not  vote is 0.0833.