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
ANSWERED
asked 2021-02-12
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?

Answers (1)

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.
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