91Ó°ÊÓ

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

Considered events:

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

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

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

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

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) =?

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

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

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

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

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

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

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

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

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

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

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

$\left|L_M\right|=500−252=248$

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

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

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