Question

A hat contains 20 names , 12 of which are female . If five names are drawn from the hat (without replacement) , what is the probability that there are at least two female names drawn ?

Answer 1

$P(x\geq 2)$ is the probability that there are at least two female names drawn

$P(x\geq 2)=1-P(x<2)=1-P(x=1)-P(x=0)$

$_{20}C_5$ is the number of ways we can choose 5 names from 20.

$P(x=1)$ is the probability that there is one female name drawn, so there are $_{12}C_1$ ways to choose 1 female name from 12, and there are $_8C_4$ ways to choose other 4 male names from 8.

$P(x=1)=\frac{_{12}C_1 \times _8 C_4}{_{20} C_5}=\frac{12\times 70}{15504}=\frac{35}{646}$

$P(x=0)$ is the probability that there is no female names drawn, so all 5 names are male, we can choose them in $_8C_5$ ways.

$P(x=0)= \frac{_8 C_5}{_{20} C_5}=\frac{56}{15504}=\frac{7}{1938}$

$P(x\geq 2)=1-\frac{35}{646}-\frac{7}{1938}=\frac{913}{969}=0.942$

Answer: 0.942