| OK I'll give it another go :) The chance of you finding your number with 50 random opens is 50%. But you don't live jusr because you find your number. You only live if everyone finds their number. And that chance is slim: it's .5^100 which is a very small number. The chance of finding your number using the chain algo is the same as the chance that your number is found before the 50th step of your chain. That probability is still 50% because you are opening at most 50 boxes of a 100, the order doesn't matter. This is why you feel nothing will change, that the order shouldn't matter. That is correct: each prisoner opens 50 boxes without any prior information. But the key is what changes for the probability of success for the group. All prisoners use the same shuffling (it's only randomized once). Everyone will succeed when walking their chains, if the longest chain is shorter than 50. That is, if there are two chains of 60,40 then the group will fail. Because some of the prisoners who are unlucky and have numbers in the 60 loop will to reach their number in 50 steps.
But if there are two chains of 50,50 or 3 chains 30,30,40 etc., then everyone will succeed. So the difference is that with random choice, YOUR chance of finding YOUR number is 50%. The chance that all 100 does so you live, is .5^100. With the chain walk, YOUR chance is still 50%, as is everyone elses. Yet the risk that someone fails is now just 70% instead 99.9999999...% (Now is no multiplication - the probabilities are not independent - the probability of success for the group is now a property of the loops rather than the product of 100 independent events) |