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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:

  1. Probability a single person would win.
  2. How many people are playing.
  3. How many times is this played (e.x. the probability anyone wins the lottery over 10 years, a.k.a many lotteries over time).
Terran
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1 Answers1

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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

$$ P(\text{at least one person wins at least one lottery}) = 1-(1-p)^{KN} $$

Brian Tung
  • 35,584
  • You got there just before me, with a better answer. The OP might like an example with a typical $p$ (perhaps $1/1,000,000$, a reasonable $K$, and $N$ for a year's worth of lotteries. That should give a small probability that no one wins. – Ethan Bolker Feb 07 '18 at 01:17
  • Thanks! I arrive at a similar equation when trying myself but it ended up being wrong. This will be very useful! – Terran Feb 07 '18 at 01:22