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While I'm reading Weird Maths: At the Edge of Infinity and Beyond by David Darling and Agnijo Banerjee, I found this part that I can't understand.

Of course, it’s extremely unlikely that you’ll win the jackpot twice. But when considering the probability that someone will, you need to multiply the odds by the number of people who play the lottery, which greatly reduces those odds, as well as the number of ways they can win the lottery twice (approximately half the square of the number of times they play the lottery). After all of this, the odds look much more reasonable that someone, somewhere will scoop the jackpot twice.

I know the odds is p/(1-p), but still don't understand why the authors did this multiplication.

If you need more text, you can find here.

Seulgi So
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1 Answers1

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I don't think working with odds instead of probabilities improves understanding in this problem, so here is an answer using probabilities.

Suppose we have $n$ people playing the lottery, with each having a probability $p$ of winning, independently of all the other players. Then the total number of winners has a Binomial distribution with parameters $n$ and $p$, so the probability that there will be exactly two winners is $$\binom{n}{2} p^2 (1-p)^{n-2}$$ Now $$\binom{n}{2} = \frac{n(n-1)}{2}$$ so for large $n$, $$\binom{n}{2} \approx \frac{n^2}{2}$$ Also, for large $n$ and small $p$, $(1-p)^{n-2} \approx 1$.

Combining these approximations, we see the probability of having exactly two winners is approximately $$\frac{n^2 p^2}{2}$$ which seems to be what the quoted text is getting at.

awkward
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