I often hear of the probability a single person wins the lottery, but someone tends to eventually (more often than not) win the lottery. So I was wondering. What math formula could I use to find the probability anyone wins the lottery given the following information:
- Probability a single person would win.
- How many people are playing.
- How many times is this played (e.x. the probability anyone wins the lottery over 10 years, a.k.a many lotteries over time).
If there are $N$ people playing, and the probability of any individual person winning is $p$, then the probability that a given person does not win is $1-p$, and the probability that none of the $N$ people win (assuming independence between the various plays) is $(1-p)^N$. So the probability that at least one person wins would be one minus that, or
$$ P(\text{at least one person wins}) = 1-(1-p)^N $$
If $N$ people play each of $K$ lotteries, then the probability that none of them win any of the lotteries is $(1-p)^{KN}$, so the desired probability would be
$$ P(\text{at least one person wins at least one lottery}) = 1-(1-p)^{KN} $$