Q.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:

Wife	Husband
	Less than  $\backslash (25,000$	More than    $\backslash )25,000$
Less than $\backslash (25,000$	$212$	$198$
More than $\backslash )25,000$	$36$	$54$

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

Using the definition of conditional probability,

a) $50.4\%$

b) $21.43\%$

c) $14.52\%$

Step 1: Given Information (part a)

The probability that the husband earns less than $$25,000$? 

Step 2: Explanation (part a)

Considered events:

$H_W\text{ - a woman is a high earner (earns more than }25,000\$\text{ ) }\backslash H_M\text{ - a man is a high earner (earns more than }25,000\$\text{ ) }$

$L_W\text{ - a woman is not a high earner. }$

$L_M\text{ - a man is not a high earner. }$

$\text{ With }\left|X\right|\text{ being the number of elements in event }X\text{, the number of couples for each combination is }$

$\begin{array}{c}\left|H_M{H}_W\right|=54\\ \left|H_M{L}_W\right|=198\\ \left|L_M{H}_W\right|=36\\ \left|L_M{L}_W\right|=212\end{array}$

Calculate, if a couple from $500$ above is randomly chosen,the probabilities.

a) P(H_M) =?

b) P(H_W/H_M) =?

c) P(HW/L_M) =?

$\text{ Break event }{H}_M\text{ into only two disjoint events }{H}_M{H}_W\text{ and }{H}_M{L}_W$

$\left|H_M\right|=\left|H_M{H}_W\right|+\left|H_M{L}_W\right|\Rightarrow \left|H_M\right|=54+198=252$

By the definition of probability on equally likely set of events.

$P\left(H_M\right)=\frac{252}{500}=0.504$

Step 3: Final Answer (part a)

The probability that the husband earns less than $$25,000$ is $50.4\%$

Step 4: Given Information (part b)

The conditional probability that the wife earns more than $$25,000$given that the husband earns more than this amount? 

Step 5: Explanation (part b)

$\text{ Now starting with the definition of conditional probability, and substituting calculated }P\left(H_M\right)$

$P(H_W\mid H_M)=\frac{P\left(H_W{H}_M\right)}{P\left(H_M\right)}=\frac{\frac{\left|H_W{H}_M\right|}{500}}{P\left(H_M\right)}=\frac{\frac{54}{500}}{\frac{252}{500}}=\frac{3}{14}\approx 21.43\mathrm{\%}$

Step 6: Final Answer (part b)

The conditional probability that the wife earns more than $$25,000$given that the husband earns more than this amount is $21.43\%$

Step 7: Given Information (part c)

The conditional probability that the wife earns more than $$25,000$given that the husband earns less than this amount? 

Step 8: Explanation (part c)

$\text{ Since with this notation }{L}_M=H_M^c$

$\left|L_M\right|=500-252=248$

$\text{ Now starting with the definition of conditional probability, and substituting calculated }\left|L_M\right|$

$P(H_W\mid L_M)=\frac{P\left(H_W{L}_M\right)}{P\left(L_M\right)}=\frac{\frac{\left|H_W{L}_M\right|}{500}}{\frac{\left|L_M\right|}{500}}=\frac{\frac{36}{500}}{\frac{248}{500}}\approx 14.52\mathrm{\%}$

Step 9: Final Answer (part c)

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