First off all, I think that if Al does not see a sample, it makes the problem a bit simpler. That is, Al just tells Bob that he (Bob) is the first person that saw 25 big fishes.
I think that the number N of scientists matters, because the probability that someone will come to see Al depends on that.
Lets call B then event the lake has 75% big fishes, S the opposite and C the event someone comes, which means that someone saw 25 fishes.
Once Al sees Bob, he updates : P(B/C)=P(B)* P(C/B)/(1/2*P(C/B)+1/2*P(C/S)). When N tends toward infinity, both P(C/B) and P(C/S) tend toward 1, and P(B/SC) tends to 1⁄2. But for small values of N, P(C/B) can be very small while P(C/S) will be quite close to 1. Then the fact that someone was chosen lowers the probability of having a lake with big fishes.
If N=infinity, then the probability of being chosen is 0, and I cannot use Bayes’ theorem.
If Charlie keeps inviting scientists until one sees 25 big fishes, then it becomes complicated, because the probability that you are invited is greater if the lake has more big fishes. It may be a bit like the sleeping beauty or the absent-minded driver problem.
First off all, I think that if Al does not see a sample, it makes the problem a bit simpler. That is, Al just tells Bob that he (Bob) is the first person that saw 25 big fishes.
I think that the number N of scientists matters, because the probability that someone will come to see Al depends on that.
Lets call B then event the lake has 75% big fishes, S the opposite and C the event someone comes, which means that someone saw 25 fishes.
Once Al sees Bob, he updates :
P(B/C)=P(B)* P(C/B)/(1/2*P(C/B)+1/2*P(C/S)).
When N tends toward infinity, both P(C/B) and P(C/S) tend toward 1, and P(B/SC) tends to 1⁄2.
But for small values of N, P(C/B) can be very small while P(C/S) will be quite close to 1.
Then the fact that someone was chosen lowers the probability of having a lake with big fishes.
If N=infinity, then the probability of being chosen is 0, and I cannot use Bayes’ theorem.
If Charlie keeps inviting scientists until one sees 25 big fishes, then it becomes complicated, because the probability that you are invited is greater if the lake has more big fishes. It may be a bit like the sleeping beauty or the absent-minded driver problem.
Edited for formatting and misspellings