Question

In a certain federal prison, it is known that 2/3 of the inmates are under 25 years of age. It is also known that 3/5 of the inmates are male and that 5/8 of the inmates are female or 25 years of age or older. What is the probability that a prisoner selected at random from this prison is female and at least 25 years old?

Answer 1

Let

U- under 25

M- male

F-female

$P(U)=\frac{2}{3}$

$P(M)=\frac{3}{5}$

$P(F\cup\overline U)=\frac{5}{8}$

$P(F\cap\overline U)$

Let

$P(M\cap\overline U)=x$

Then

$P(M\cap{U})=P(M)-P(M\cap \overline U)=\frac{3}{5}-x$

Similarly

$P(F\cap U)=P(U)-P(M\cap U)=\frac{2}{3}-\frac{3}{5}+x=\frac{1}{15}+x$

$P(F\cap\overline U)= P(F)-P(F\cap U)$

$=\frac{5}{8}-x-(\frac{1}{15}-x-)=\frac{67}{120}-2x$

Now

$P(M)+P(F)=1$

$\frac{3}{5}+\frac{5}{8}-x=1$

Solving for x, we have,

$x=\frac{9}{40}$

Thus,

$P(F\cap\overline U)=\frac{67}{120}-2(\frac{9}{40})=\frac{13}{120}$