Question

In a certain federal prison it is known that 2/3 of the inmates are under 25 yrs of age. Its is also known that 3/5 of the inmates are males and that 5/8of the inmates are females 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 $A$ denotes the event "inmate is under 25 years of age".

Let $M$ denotes the event "inmate is a male".

Let $F$ denotes the event "inmate is a female".

Given $P(A)=\dfrac{2}{3}, P(M)=\dfrac{3}{5}, P(F\cup A')=\dfrac{5}{8}.$

$$P(A')=1-P(A)=1-\dfrac{2}{3}=\dfrac{1}{3}$$ $$P(F)=1-P(M)=1-\dfrac{3}{5}=\dfrac{2}{5}$$ $$P(F\cap A')=P(F)+P(A')-P(F\cup A')$$ $$P(F\cap A')=\dfrac{2}{5}+\dfrac{1}{3}-\dfrac{5}{8}=\dfrac{48+40-75}{120}$$ $$=\dfrac{13}{120}$$