Question

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 $Po(4)$

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