Question

s Consider a cub couto of American and Fre 40 and fr endh members Prob males ond Prabl Fenales fiven het neubers are Freuch. Find the Probabilly of Amen can femaleふ ei tin mentbers h e

Answer 1

The following probabilities are given

$\fn_jvn P\left ( \textup{Males} \right )=\frac{13}{18}$. Hence $\fn_jvn P\left ( \textup{Females} \right )=1-\frac{13}{18}=\frac{5}{18}$

$\fn_jvn P\left ( \textup{Males}|\textup{American} \right )=\frac{10}{11}$. Hence $\fn_jvn P\left ( \textup{Females}|\textup{American} \right )=1-\frac{10}{11}=\frac{1}{11}$

$\fn_jvn P\left ( \textup{Females}|\textup{French} \right )=\frac{4}{7}$

Using total probability theorem,

$\fn_jvn P\left ( \textup{Females}|\textup{American} \right )P\left ( \textup{American} \right )+P\left ( \textup{Females}|\textup{French} \right )P\left ( \textup{French} \right )=P\left ( \textup{Females}\right )\\ \frac{1}{11}P\left ( \textup{American} \right )+\frac{4}{7}\left (1-P\left ( \textup{American} \right ) \right )=\frac{5}{18}\\ P\left ( \textup{American} \right )=\frac{4/7-5/18}{4/7-1/11}\\ P\left ( \textup{American} \right )=0.61111111$

Thus the probability (using conditional probability),

$\fn_jvn P\left ( \textup{Females}\cap \textup{American} \right )=P\left ( \textup{Females}|\textup{American} \right )P\left (\textup{American} \right )=\frac{1}{11}(0.61111111)\\ {\color{Blue} P\left ( \textup{Females}\cap \textup{American} \right )=0.0556}$