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.
Brainy_Gal t1_itqqzh7 wrote
Reply to How does one figure out what probability is most relevant when deciding how probable something is? by eth_trader_12
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.