federal prison

[logistic_guy]

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

 

[logistic_guy]

Let us first understand some of the tricks in the problem. The statement \(\displaystyle 25\) years of age or older means that the age is at least \(\displaystyle 25\).

Or

The age \(\displaystyle \geq 25\).

If we let \(\displaystyle F\) denote female and \(\displaystyle A\) denote at least \(\displaystyle 25\) years of age, then they want us to the find:

\(\displaystyle P(F \cap A)\)

The beauty about this problem is that both probabilities \(\displaystyle P(F)\) and \(\displaystyle P(A)\) are not given!

 

[logistic_guy]

If \(\displaystyle 2/3\) of the inmates are under \(\displaystyle 25\) years of age, then the probability the inmates are \(\displaystyle 25\) or older is:

\(\displaystyle P(A) = 1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{1}{3}\)

 

[logistic_guy]

If \(\displaystyle 3/5\) of the inmates are male, then \(\displaystyle 2/5\) are female. That is:

\(\displaystyle P(F) = \frac{2}{5}\)

 

[logistic_guy]

\(\displaystyle P(F \cup A) = \frac{5}{8}\) was given.

Then,

\(\displaystyle P(F \cup A) = P(F) + P(A) - P(F \cap A)\)

Plug in numbers.

\(\displaystyle \frac{5}{8} = \frac{2}{5} + \frac{1}{3} - P(F \cap A)\)

This gives:

What is the probability that a prisoner selected at random \(\displaystyle \text{f}\)rom this prison is female and at least \(\displaystyle 25\) years old?

\(\displaystyle P(F \cap A) = \frac{13}{120} = \textcolor{blue}{0.1083}\)

 

[Agent Smith]

Can I use 120 total prisoners?

40 are under 25 years of age

75 are female or 25 years/younger

72 are male. 48 are female

75 - 48 = 27 are only younger than 25

40 - 27 = 13 are both younger than 25 and female

Probability = 13/120