An intramural basketball team is to be chosen randomly from 15 boys and 12 girls. The team has ten slots. You want to know the probability that eight of the players will be boys. What is the group of interest and the sample?

An intramural basketball team is to be chosen randomly from 15 boys and 12 girls. The team has ten slots. To know the probability that eight of the players will be boys and the group of interest and the sample.

Let X the number of boys players on the team of eight players. The boys are the group of interest and the sample size X takes on the values $0,1,2,3,4,5,6,7,8,9,10$.

Here X follows a hypergeometric distribution with $K=15$boys in population $N=27$, where here are $k=8$successes from the sample size $n=10$.

The probability mass function of hypergeometric distribution of X is written as

$P(X=k)=\frac{\left({}^{K}C_k\right)\left({}^{N-K}C_{n-k}\right)}{\left({}^{N}C_n\right)}$

Therefore the required probability of eight of the players will be boys out off ten slots team is determined as:

$\begin{array}{r}P(X=8)=\frac{\left({}^{15}{C}_8\right)\left({}^{27-15}{C}_{10-8}\right)}{\left({}^{27}{C}_{10}\right)}\\ =\frac{6435\times 66}{8436285}\\ =0.0503\end{array}$

Therefore the required probability of eight of the players will be boys out off ten slots team is $0.0503$.