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:
\(T\) = inmate under 25 years old
\(\overline{T}\) = inmate 25 years or older
\(M\) = male
\(\overline{M}\)= female

You are given:
\(P(T)=2/3\)
\(P(M)=3/5\)
\(P(\overline{M} \cup \overline{T})=5/8\)

You need:
\(P(\overline{M} \cap \overline{T})\)

Thus:
\(P(\overline{M} \cup \overline{T})=P(\overline{M})+P(\overline{T})-P(\overline{M} \cap \overline{T})\)
\(\implies P(\overline{M}\cap \overline{T})=P(\overline{M})+P(\overline{T})-P(\overline{M}\cup \overline{T})\)
\(\implies P(\overline{M}\cap \overline{T})=[1-P(M)]+[1-P(T)]-P(\overline{M}\cup \overline{T})\)
\(\implies P(\overline{M}\cap \overline{T})=[1-3/5]+[1-2/3]-5/8\)

\(\therefore P(\overline{M} \cap \overline{T})=13/120 \approx 0.1083\)

Answer 2

THANKS! I was close but somethings weren't matching up in my head.

Answer 3

=]