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Problem 100: A shipment of 120 burglar alarms contains 5 that are defective. If 3 of these alarms are randomly selected and shipped to a customer, find the probability that the customer will get one bad unit by using
a. the formula for the hypergeometric distribution;
b. the formula for the binomial distribution as an approximation.

Problem 100: A shipment of 120 burglar alarms contains 5 that are defective. If 3 of these alarms are randomly selected and shipped to a customer, find the probability that the customer will get one bad unit by using
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 $x=1,n=3,a=5, N=120$
Hypergeometric distribution, $h(x;n,a,N)=\frac{\left(\begin{array}{c}a\\ x\end{array}\right)\left(\begin{array}{c}N-a\\ n-x\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)}$
$P(x=1) = h(1;3,5,120)=\frac{\left(\begin{array}{c}5\\ 1\end{array}\right)\left(\begin{array}{c}120-5\\ 3-1\end{array}\right)}{\left(\begin{array}{c}120\\ 3\end{array}\right)} = 0.1167.$

b. the formula for the binomial distribution as an approximation
Given that $x=1,n=3,p = \frac{5}{120}$
$P(x= 1) = b(1;3,\frac{5}{120})$
$b(x;n,p)=\left(\begin{array}{c}n\\ x\end{array}\right){p}^{x}(1-p)^{n-x}$
$P(x= 1) = b(1;3,\frac{5}{120}) = \left(\begin{array}{c}3\\ 1\end{array}\right){\frac{5}{120}}^{1}(1-\frac{5}{120})^{3-1} = 0.1148$

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