I was messing around with PMF and thought about what'd happen if the probability of getting a success was nearly impossible, and the amount of tries approached infinity (and we only wanted 1 success). $$f(n,p,k)=\frac{n!p^k(1-p)^{n-k}}{k!(n-k)!}\hspace{0.1cm}\Bigg|\hspace{0.3cm} n=\text{number of trials} \\ p=\text{probability of getting a success in one trial},\hspace{0.5cm}k=\text{number of successes desired} \\[5pt] f\left(\lim_{n\to\infty},\lim_{p\to 0},1\right)=\frac{1}{e}$$ Now why exactly does $e$ show up here? does it have any meaning? is the limit correct? I'm very confused and thanks in advance
2026-02-23 17:21:01.1771867261
Euler's constant in probability mass function?
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