3.21 - A First Course in Probability

Question

A total of 500 married working couples were polled about their annual salaries, with the following information resulting: 

For instance, in 36 of the couples, the wife earned more and the husband earned less than \( 25,000. If one of the couples is randomly chosen, what is

(a) the probability that the husband earns less than \) 25,000 ?

(b) the conditional probability that the wife earns more than \( 25,000 given that the husband earns more than this amount?

(c) the conditional probability that the wife earns more than \) 25,000 given that the husband earns less than this amount?

Step-by-Step Solution

Answer

a. The probability that the husband earns less than $ 25,000 is $0.496$.

b. The conditional probability that the wife earns more than $ 25,000 given that the husband earns more than this amount is $0.214$.

c. The conditional probability that the wife earns more than $ 25,000 given that the husband earns less than this amount is $0.145$. 

Step 1 : Given Information

$500\text{ married couples were surveyed about their annual earnings. }$

Step 2 : Formula Used

$\text{ Probability of an event }=\frac{\text{ Number of favorable outcomes }}{\text{ Number of total outcomes }}$

Step 3 : Calculation (Part a)

$\text{ Couples in which the husband earns less than }\$25,000=212+36=248$

 The probability that the husband earns less than $ 25,000$=\frac{248}{500}=0.496$

Step 4 : Calculation (Part b)

$\text{ Couples in which husbands more than }\$25,000=198+54=252$

$\text{ Couples in which the wife earns more than }\$25,000\text{ given the husband earns more than }\$25,000=54$

$\text{ Probability that the wife earns more than }\$25,000\text{ given the husband earns more than this amount }$$=\frac{54}{252}=\frac{3}{14}=0.214$

Step 5 : Calculation (Part c)

$\text{ Couples in which husbands less than }\$25,000=212+36=248$

$\text{ Couples in whichthe wife earns more than }\$25,000\text{ given the husband earns less than this amount }=36$

$\text{ Probability that the wife earns more than }\$25,000\text{ given the husband earns less than this amount }$$=\frac{36}{248}=\frac{9}{62}=0.145$