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Imagine we have a lottery, with chance of winning 1 in 1000 (1/1000). I have made computer simulations in order to find confidence levels for winning. At 1000 bought lottery tickets, the confidence of winning is 64.1% and 2000 bought lottery tickets the confidence of winning is 87.1%

By changing the chance of winning to 1/X, I have verified that there is always 64.1% confidence at X bought lottery tickets, and 87.1% confidence at 2X bought tickets.

Clearly, there must be a probability density function that describe these kind of events but I have failed at finding such in all of my textbooks at home. I was thinking about the Poisson distribution, but that just tell us the distribution of the number of wins for a certain number of bought lottery tickets.

I am only interested if a person wins or not, not how many wins they get. So, should I integrate the Poisson distribution from 1 to infinity? And if Poisson is the correct distribution, how to I figure out the "mean" from the chance of winning?