Answer to Question #283261 in Statistics and Probability for Rapshowboy

Question

1. The plant sent 2,000 thermometers to the pharmacy warehouse. The probability of damage to each thermometer along the way is 0.002. What is the probability that exactly 3 damaged thermometers will arrive at the pharmacy warehouse

Answer 1

When the value of $n$ in a binomial distribution is large and the value of $p$  is very small, the binomial distribution can be approximated by a Poisson distribution. If $n > 20$ and $np < 5$ or $nq < 5$ then the Poisson is a good approximation.

Given

$n=2000>20,$

$p=0.002, np=2000(0.002)=4<5$

The Poisson is a good approximation $Po(4)$

$$P(X=3)\approx\dfrac{e^{-4}(4)^3}{3!}\approx0.195367$$