Question

Forty percent of prison inmates were unemployed when they entered prison. If 5 inmates were randomly selected, find these probabilities.

a. Exactly 3 were unemployed.
b. At most 4 were unemployed.
c. At least 3 were unemployed.
d. Fewer than 2 were unemployed.

Answer 1

A. 0.2304

B. 0.98976

C. 0.91296

D. 0.33696

Explanation:

Forty percent of prison inmates were unemployed when they entered prison, then

`p=0.4\\ \\q=1-p=1-0.4=0.6`

Find all needed probabilities:

A. Exactly 3 were unemployed

`C^5_3p^3q^(5-3)=(5!)/(3!(5-3)!)p^3q^2=10(0.4)^3(0.6)^2=0.2304`

B. At most 4 were unemployed

`C^5_0p^0q^(5-0)+C^5_1p^13q^(5-1)+C^5_2p^2q^(5-2)+C^5_3p^3q^(5-3)+C^5_4p^4q^(5-4)=1-C^5_5p^5q^(5-5)=1-(5!)/(5!(5-5)!)p^5q^0=1-1\cdot (0.4)^5\cdot 1=1-0.01024=0.98976`

C. At least 3 were unemployed

`C^5_0p^0q^(5-0)+C^5_1p^13q^(5-1)+C^5_2p^2q^(5-2)+C^5_3p^3q^(5-3)=1-C^5_5p^5q^(5-5)-C^5_4p^4q^(5-4)=1-1\cdot p^5q^0-5\cdot (0.4)^4(0.6)^1=1-0.01024-0.0768=0.91296`

D. Fewer than 2 were unemployed

`C^5_0p^0q^(5-0)+C^5_1p^13q^(5-1)=1\cdot (0.6)^5+5\cdot (0.4)(0.6)^4=0.0776+0.2592=0.33696`