# A famous NBA player appears at a local hot spot an average once every month. What is the probability that he will make an appearce at this same local that hot spot more than 2 times in a three month span?

A famous NBA player appears at a local hot spot an average once every month. What is the probability that he will make an appearce at this same local that hot spot more than 2 times in a three month span?
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Arham Warner
Let X be number of times a famous NBA player appears at a local hot spot.
Since this is a rare event, X follows Poisson distribution with mean 1. That is, $\lambda =1$ per month.
For three months $\lambda =1\cdot 3=3.$
If X is the Poisson random variable, then the probability mass function of X is
$P\left(X=x\right)=\frac{{e}^{-\lambda }{\lambda }^{x}}{x!},x=0,1,2$,.......
Then, the probability that he will make an appearance at the local hot spot more than 2 times is
$P\left(X>2\right)=1-P\left(X\le 2\right)$
$=1\left\{P\left(X=0\right)+P\left(X=1\right)+P\left(X=2\right)\right\}$
$=1-\left\{\frac{{e}^{-3}{3}^{0}}{0!}+\frac{{e}^{-3}{3}^{1}}{1!}+\frac{{e}^{-3}{3}^{2}}{2!}\right\}$
$=1-\left\{0.0498+0.1494+0.2240\right\}$
$=1-0.4232$
$=0.5768$
Thus, the probability that he will make an appearance at the local hot spot more than 2 times is 0.5768.