Let’s go for a large, finite, case. Because otherwise my brain will explode.
Question 1: for any large, finite number of scientists Bob should defer MOSTLY to Alice.
First lets look at Alice; In any large finite number of scientists there is a small finite chance that NO scientist will get that result.
This chance is larger in the case where 75% of the fish are big.
Thus, upon finding that a scientist HAS encountered 25 fish, Alice must adjust her probability slightly towards 25% big fish.
Bob has also received several new pieces of information.
*He was the first to find 25 big fish. P[first25|found25] approaches 1/P[found25] as you increase the number of scientists. This information almost entirely cancels out the information he already had.
*All the information Alice had. This information therefore tips the scales.
Bob’s final probability will be the same as Alice’s.
Question two is N/A
I will answer question three in a reply to this to try and avoid a massive wall of text.
Question 3:
lateral answer: in the symmetrical variant the issue of “how many people are being given other people to meet, and is this entire thing just a weird trick” begins to rise.
In fact, the probability of it being a weird trick is going to overshadow almost any other attempt at analysis. The first person to get 25 happens to be a person who is told they will meet someone who got 75, and the person who was told they would meet the first person to get 25 happens to get 75? Massively improbable.
However, if it is not a trick, the probability is significantly in favour of it being 75% still. Alice isn’t talking to Bob due to the fact she got 75, she’s talking to Bob due to the fact he got the first 25. Otherwise Bob would most likely have ended up talking to someone else.
The proper response at this point for both Alice and Bob is to simply decide that it is overwhelming probable that Charlie is messing with them.
I can produce similar variants which don’t have this issue, and they come out to 50:50. These include: Everyone is told that the first person to get 25 will meet the first person to get 75.
Okay, qualitative analysis without calculations:
Let’s go for a large, finite, case. Because otherwise my brain will explode.
Question 1: for any large, finite number of scientists Bob should defer MOSTLY to Alice.
First lets look at Alice; In any large finite number of scientists there is a small finite chance that NO scientist will get that result. This chance is larger in the case where 75% of the fish are big. Thus, upon finding that a scientist HAS encountered 25 fish, Alice must adjust her probability slightly towards 25% big fish.
Bob has also received several new pieces of information.
*He was the first to find 25 big fish. P[first25|found25] approaches 1/P[found25] as you increase the number of scientists. This information almost entirely cancels out the information he already had.
*All the information Alice had. This information therefore tips the scales.
Bob’s final probability will be the same as Alice’s.
Question two is N/A I will answer question three in a reply to this to try and avoid a massive wall of text.
Question 3: lateral answer: in the symmetrical variant the issue of “how many people are being given other people to meet, and is this entire thing just a weird trick” begins to rise.
In fact, the probability of it being a weird trick is going to overshadow almost any other attempt at analysis. The first person to get 25 happens to be a person who is told they will meet someone who got 75, and the person who was told they would meet the first person to get 25 happens to get 75? Massively improbable.
However, if it is not a trick, the probability is significantly in favour of it being 75% still. Alice isn’t talking to Bob due to the fact she got 75, she’s talking to Bob due to the fact he got the first 25. Otherwise Bob would most likely have ended up talking to someone else.
The proper response at this point for both Alice and Bob is to simply decide that it is overwhelming probable that Charlie is messing with them.
I can produce similar variants which don’t have this issue, and they come out to 50:50. These include: Everyone is told that the first person to get 25 will meet the first person to get 75.