Brainy_Gal t1_itqqzh7 wrote on October 25, 2022 at 4:47 PM

Good question! To answer it, I'll pose another one: why don't we take it as evidence the lottery was rigged every single time someone wins (i.e. not twice in a row but just once)? In other words why is: Given the lottery isn't rigged, the probability that SOMEONE will win is (close to) 1 the relevant statement, instead of: Given the lottery isn't rigged, the probability that JULIET will win is very small. Clearly if we use the latter approach, we will (almost) always find "evidence" the lottery is rigged even when it isn't, so this can't be correct. And the reason why is that we've formulated the probability statement AFTER we know Juliet won.

So, with someone winning twice in a row; the relevant statement is: given the lottery isn't rigged, what is the probability that SOMEONE will win twice in a row? Assuming that's reasonably high, we can't use this as evidence the lottery is rigged. But let's assume it's low; you may rightly ask: why isn't the relevant question: given the lottery isn't rigged, what is the probability that someone will win twice (at any two time points)? Why are we assigning special weight to the fact that it's twice in a row? And here is where there is really no good answer. Yes it's true we formulated the former probability statement before knowing someone actually did win twice in a row. Or did we? What if Juliet had not won back-to-back, but twice within the space of a week? Would we still insist this was evidence of the lottery being rigged? These are the questions that keep statisticians awake at night.

eth_trader_12 OP t1_itqty1l wrote on October 25, 2022 at 5:06 PM

I think the relevant question even in your example of just one lottery is to consider the probability that Juliet winning is small. And there's no problem in that. The probability of Juliet winning given that it was rigged is 1. The probability of Juliet winning normally is very small.

But that's fine. That's the probability of Juliet winning given chance, not the probability of chance given Juliet winning. In order to arrive at the second, we need to look at prior probabilities. Given that the rate at which lotteries is rigged is probably less than the chance of Juliet winning, one would still conclude Juliet won fairly.

An example of something that makes this clear is imagining that every other lottery is rigged. If every other lottery is rigged, would it make sense to look at the probability of merely "someone" winning the lottery. Clearly not. Since the probability of "someone" winning the lottery is very high if not 1 depending on the lottery. But does that mean the probability of it occurring by chance is close to 1? No. Because every other lottery is still rigged.