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Problem 101: Among the 300 employees of a company, 240 are union members, while the others are not, If 8 of the employees are chosen to serve on the administrative committee, find the probability that 5 of them will be union member while the others are not.
a. the formula for the hypergeometric distribution;
b. the formula for the binomial distribution as an approximation.

Problem 101: Among the 300 employees of a company, 240 are union members, while the others are not, If 8 of the employees are chosen to serve on the administrative committee, find the probability that 5 of them will be union member while the others are not.
a. the formula for the hypergeometric distribution;
b. the formula for the binomial distribution as an approximation.


Solution:


a. the formula for the hypergeometric distribution
Given that $x=5,n=8,a=240, N=300$
Hypergeometric distribution, $h(x;n,a,N)=\frac{\left(\begin{array}{c}a\\ x\end{array}\right)\left(\begin{array}{c}N-a\\ n-x\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)}$
$P(x=5) = h(5;8,240,300)=\frac{\left(\begin{array}{c}240\\ 5\end{array}\right)\left(\begin{array}{c}300-240\\ 8-5\end{array}\right)}{\left(\begin{array}{c}300\\ 8\end{array}\right)} = 0.1470.$

b. the formula for the binomial distribution as an approximation?
Given that $x=5,n=8,p = \frac{240}{300}$
$P(x= 5) = b(5;8,\frac{240}{300})$
$b(x;n,p)=\left(\begin{array}{c}n\\ x\end{array}\right){p}^{x}(1-p)^{n-x}$
$P(x= 5) = b(5;8,\frac{240}{300}) = \left(\begin{array}{c}8\\ 5\end{array}\right){\frac{240}{300}}^{5}(1-\frac{240}{300})^{8-5} = 0.1468$

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