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?

Short answer

Using the definition of conditional probability,

a)$50.4\%$

b)$21.43\%$

c)$14.52\%$

Step-by-step solution

Given Information (part a)

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

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.

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

Final Answer (part a)

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

Given Information (part b)

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

Explanation (part b)

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

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

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$21.43\%$

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)

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

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

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

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

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