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The waiting time paradox: why is my bus always late? (2018)

(jakevdp.github.io)
219 points skadamat | 1 comments | 20 Aug 24 13:55 UTC | HN request time: 0.229s | source
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xnorswap ◴[20 Aug 24 15:50 UTC] No.41301135[source]▶
My favourite corollary of this is that even if you win the lottery jackpot, then you win less than the average lottery winner.

Average Jackpot prize is JackpotPool/Average winners.

Average Jackpot prize given you win is JackpotPool/(1+Average winners).

The number of expected other winners on the date you win is the same as the average number of winners. Your winning ticket doesn't affect the average number of winners.

This is similar to the classroom paradox where there are more winners when the prize is poorly split, so the average observed jackpot prize is less than the average jackpot prize averaged over events.

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1. GuB-42 ◴[20 Aug 24 20:11 UTC] No.41303568[source]▶
That's true if you are cheating, for example by knowing the numbers in advance, guaranteeing a win. The cheater is the "+1" in your argument, an extra player with a 100% win rate.

But if you are not, and pick a random time where you win, on average, you will win as much as the average lottery winner.

For the classroom paradox to work, you have to take the average prize per draw after splitting, not the average prize per winner.

For example, if there are 9 winners in the first draw and 1 in the second, then there are 5 winners on average, so the average prize is 1/5. If you are one of the winners, there is 9/10 chance you are among the 9 and only win 1/9, which is less than average, but there is also 1/10 change of winning full prize, which is much better than average. If you take a weighed average of these (9/10*1/9+1/10*1) you get 1/5, back to the average prize. The average individual prize per draw is (1/9+1)/2=5/9, but it is kind of a meaningless number.

Another way to see it is that most of the times, you will win less than average, but the few times you win more, then you will win big. But isn't it what lotteries are all about?