# The Mathematical Ninja and the Poisson Distribution | Colin

“What are the ch…”

“About 11.7%,” said the Mathematical Ninja. “Assuming $X$ is drawn from a Poisson distribution with a mean of 9 and we want the probability that $X=7$.”

“That’s a fair assumption, sensei,” pointed out the student, “given that that’s what the sodding question says.” A wiser student may not have pushed their luck. “So, where does 12% come from?”

“It’s simple,” said the Mathematical Ninja. “You know, of course, that $P(X=n) = e^{-\lambda} \frac{\lambda^n}{n!}$?”

“But of course! I remember it because it’s the $n$th term of a Maclaurin series for $e^{-\lambda} e^{x}$…