Question

# 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?

Modeling
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?

2021-02-13
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, $$\displaystyle\lambda={1}$$ per month.
For three months $$\displaystyle\lambda={1}\cdot{3}={3}.$$
If X is the Poisson random variable, then the probability mass function of X is
$$\displaystyle{P}{\left({X}={x}\right)}=\frac{{{e}^{-\lambda}\lambda^{x}}}{{{x}!}},{x}={0},{1},{2}$$,.......
Then, the probability that he will make an appearance at the local hot spot more than 2 times is
$$\displaystyle{P}{\left({X}>{2}\right)}={1}-{P}{\left({X}\le{2}\right)}$$
$$\displaystyle={1}{\left\lbrace{P}{\left({X}={0}\right)}+{P}{\left({X}={1}\right)}+{P}{\left({X}={2}\right)}\right\rbrace}$$
$$\displaystyle={1}-{\left\lbrace\frac{{{e}^{{-{3}}}{3}^{0}}}{{{0}!}}+\frac{{{e}^{{-{3}}}{3}^{1}}}{{{1}!}}+\frac{{{e}^{{-{3}}}{3}^{2}}}{{{2}!}}\right\rbrace}$$
$$\displaystyle={1}-{\left\lbrace{0.0498}+{0.1494}+{0.2240}\right\rbrace}$$
$$\displaystyle={1}-{0.4232}$$
$$\displaystyle={0.5768}$$
Thus, the probability that he will make an appearance at the local hot spot more than 2 times is 0.5768.