Answer to Question #287897 in Statistics and Probability for Sanjana

Question

The plant sent 5,000 thermometers to the pharmacy warehouse. The probability that

the product will be damaged on the way is 0.0002. What is the likelihood that three 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 Poisson approximation to Binomial can be used. For this approximation to be applied, the following conditions must be met.

$1)\space n\gt20\\ 2)\space np\lt5 \space or\space n(1-p)\lt5$

Here,

$n=5000\gt20\\ p=0.0002\\ np=(5000\times0.0002)=1\lt5$

Therefore, the Poisson distribution is a good approximation.

The Poisson distribution to be used has parameter $\lambda=np=1$.

We determine the probability $P(X=3)$=${e^{-1}1^3\over 3!}={e^{-1}\times1\over 3!}=0.06131324$

Thus, the likelihood that three damaged thermometers will arrive at the pharmacy warehouse is 0.06131324.