# 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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