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.

How the chance nobody wins depends on number of tickets sold
Number of tickets sold % Chance nobody wins
$N$ 37%
$2N$ 13%
$3N$ 5%
$4N$ 2%
$5N$ 1%
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