Will it be a rollover?
Suppose there are $aN$ lottery tickets sold, each with a chance $1/N$ of winning.
Then each has a chance $1 - 1/N$ of losing, and the chance that they all lose is
$$\left(1-\frac{1}{N}\right)^{Na} \approx e^{-a},$$
where $e=2.718$ is the base of natural logarithms, and is also the limit of $(1 + 1/x)^x$ as $x$ gets large. For $a = 1,2,3,4,5$ this gives the results in the Table.
| Number of tickets sold | % Chance nobody wins |
|---|---|
| $N$ | 37% |
| $2N$ | 13% |
| $3N$ | 5% |
| $4N$ | 2% |
| $5N$ | 1% |
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