Question

In math club there are 7 girls and 5 boys. A team of 6 students must be formed to coMpete In math contest.let x be the number of boys in the team.

1. What are the posible value of x?

2. Construct a probability distribution for the random variable.

Answer 1

Let's denote B - boys, G - girls.

Sample space S is all possible outcomes.

Let $X$ be the random variables representing the number of number of boys in the team.

1. The possible values of $x$ are $0,1,2,3,4,5.$

2. There are $\dbinom{7+5}{6}=924$ possible outcomes.

$$P(X=0)=\dfrac{\dbinom{5}{0}\dbinom{7}{6}}{\dbinom{12}{6}}=\dfrac{1(7)}{924}=\dfrac{1}{132}$$

$$P(X=1)=\dfrac{\dbinom{5}{1}\dbinom{7}{5}}{\dbinom{12}{6}}=\dfrac{5(21)}{924}=\dfrac{5}{44}$$

$$P(X=2)=\dfrac{\dbinom{5}{2}\dbinom{7}{4}}{\dbinom{12}{6}}=\dfrac{10(35)}{924}=\dfrac{25}{66}$$

$$P(X=3)=\dfrac{\dbinom{5}{3}\dbinom{7}{3}}{\dbinom{12}{6}}=\dfrac{10(35)}{924}=\dfrac{25}{66}$$

$$P(X=4)=\dfrac{\dbinom{5}{4}\dbinom{7}{2}}{\dbinom{12}{6}}=\dfrac{5(21)}{924}=\dfrac{5}{44}$$

$$P(X=5)=\dfrac{\dbinom{5}{5}\dbinom{7}{1}}{\dbinom{12}{6}}=\dfrac{1(7)}{924}=\dfrac{1}{132}$$

Construct the probability distribution of the random variable

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline p(x) & 1/132 & 5/44 & 25/66 & 25/66 & 5/44 & 1/132 \end{array}$$