# a) Find the value of p(10) by using hypergeometric probability

a) Find the value of p(10) by using hypergeometric probability distribution.
b) Find the value of p(10) by using binomial probability distribution.
Determine the value for p(10) obtained from the binomial distribution is a good approximation to that obtained using the hypergeometric distribution when the N large enough.
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a) Calculation:
A hypergeometric probability distribution of random variable Y is,

Let Y is defined as the number of defectives in a random sample of size 20. A lot of 100 industrial products contains 40 defectives. That is, $r=40,N=100,n=20$.
The value of p(10) is,
$P\left(Y=10\right)=\frac{\left(\begin{array}{c}40\\ 10\end{array}\right)\left(\begin{array}{c}100-40\\ 20-10\end{array}\right)}{\left(\begin{array}{c}100\\ 20\end{array}\right)}$
$=0.11922$
Hence, the value of p(10) by using hypergeometric probability distribution is 0.1192.
b) Calculation:
A random variable Y is a binomial distribution based on n trails with success probability p if and only if,

Let Y is defined as the number of defectives. A random sample of size 20 with a lot of 100 industrial products contains 40 defectives. That is, $n=20,p=0.4\left(=\frac{40}{100}\right)$,
The value of p(10) is,
$P\left(Y=10\right)=\left(\begin{array}{c}20\\ 10\end{array}\right)\left(0.4{\right)}^{10}\left(0.6{\right)}^{20-10}$
$=0.117$
Hence, the value of p(10) by using binomial probability distribution is 0.117.
It is clear that the value of p(10) by using binomial approximation and using hypergeometric distribution are almost same this indicates that when N is large the binomial distribution is a good approximation to hypergeometric distribution.