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Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople).

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ORIGINAL: GroupW

Interesting puzzle. Just thinking it through on the fly, if the islanders can see each other, they certainly realize that there are more than one person with blue eyes, so there is no new information that the visitor brings and one might assume that on this basis nothing changes. One could know the probability of having either brown or blue eyes, but could never ascertain the precise color of his own. The world - or at least the island - is safe.

On the other hand, the math fundamentally changes if you look at things day by day recursively.

Day 1 - If there were one person with blue eyes, and all other eyes were brown, then the person with the blue eyes would realize that he's the person the visitor is observing (there being no other visible blue eyed people). That person would go commit suicide.

Day 2 - If there were two blue eyed people, and noone committed suicide on day 2 (the day after the visitor opened his big mouth), the second person would realize that there was one other person with blue eyes and no others. That person would then go commit suicide along with the first one who would have come to the exact same conclusion simultaneously.

Day 3 - If on day three there were three blue eyed people and no one had as yet committed suicide, then there would be three blue eyed people who all come to the simultaneous conclusion that they are blue eyed and then all three would nip off and shoot themselves.

Each following day - That process would continue to the 101st day at which point all the blue eyed people would be dead.

The next day, noone would would commit suicide since all the remaining blue eyed folks would already be dead. At that point, the rest of the tribe realizes that they have brown eyes and follow in their blue eyed brothers footsteps. On the 103rd day, all the islanders are dead.

It's a good example of a path-dependent process or conditional probability. If you look at the situation at a single point in time and try to infer the result from that, you come to the wrong conclusion. If you break the problem down by point in time and follow the process to a logical conclusion, then you get to the unfortunate but correct answer that they all die.

We deal with this kind of phenomenon in the finance world, particularly in mortgages. If we say that the probability of a loan defaulting is 10%, then I would expect 10% of the mortgages in a pool to default. If, however, I say that the probability of a mortgage defaulting is 10% and apply that logic sequentially over time, there is a survivorship bias that occurs and actually less than 10% of the pool goes into default. The probability that I default in the next month is 10%, but that's conditional on the fact that I have not yet defaulted. It's a similar process at work here.

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On the other hand, the math fundamentally changes if you look at things day by day recursively.

Day 1 - If there were one person with blue eyes, and all other eyes were brown, then the person with the blue eyes would realize that he's the person the visitor is observing (there being no other visible blue eyed people). That person would go commit suicide.

Day 2 - If there were two blue eyed people, and noone committed suicide on day 2 (the day after the visitor opened his big mouth), the second person would realize that there was one other person with blue eyes and no others. That person would then go commit suicide along with the first one who would have come to the exact same conclusion simultaneously.

Day 3 - If on day three there were three blue eyed people and no one had as yet committed suicide, then there would be three blue eyed people who all come to the simultaneous conclusion that they are blue eyed and then all three would nip off and shoot themselves.

Each following day - That process would continue to the 101st day at which point all the blue eyed people would be dead.

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ORIGINAL: allisonbrett

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Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople).

"initially aware? When is this statistic made known?

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ORIGINAL: GroupW

Can it be on the 100th day if they commit suicide on the following day? The earliest would be the 101st day, no? Then, you would need at least 1 suicide free day before the brown eyed people can be certain that the blue eyed people are all dead, at which point there's a one day delay before they are dead.

Where did I mess up?

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ORIGINAL: DaveW

Unless I am missing a point, I do not see why any blue eyed natives would off themselves as the visitor did not mention any numbers. His comment about seeing others with blue eyes merely confirms what each one already knows, that most have brown eyes but some have blue.

No specific numbers = no one has to off themselves.

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ORIGINAL: GroupW
That's the catch. As time goes by, you CAN actually determine the specific numbers.

Day 1 - If there were only 1 blue eyed person, the one blue eyed person would look around and see only 999 brown eyed people plus the visitor. He would automatically know that he's the only blue-eyed dude in town and thus be compelled to commit suicide.

Day 2 - If there were only 2 blue eyed people on the island, the two blue eyed people would look around and see 1 other blue eyed person plus 998 brown eyed people. As long as no one committed suicide on the previous day, he would be able to deduce that he had blue eyes. (If no one committed suicide the prior day, then there must have been at least 1 other blue eyed person on that day. Noone has committed suicide, so he can also look around and see that there can be no more than 2 total blue eyed persons, including himself. He can only see one of them, so he must be the only other blue eyed person. Hence, he must die.)

Day 3 - If there were only 3 blue eyed people on the island, each of the three would look around and see 2 blue eyed people plus 997 brown eyed people. As long as no one committed suicide the prior 2 days, he can be assured that that he is one of 3 people that has blue eyes.

That process will continue reliably for the next 100 days. When the suicides stop, all the brown eyed people can be assured they have brown eyes.

Is that clearer?

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ORIGINAL: jhuperetes

How would the first blue eyed person discover that s/he is blue eyed in your scenario?

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Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople).

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"highly logical" means that any conclusion that can logically deduced from the information and observations available to an islander, will automatically be known to that islander.

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ORIGINAL: DaveW

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ORIGINAL: GroupW
That's the catch. As time goes by, you CAN actually determine the specific numbers.

Day 1 - If there were only 1 blue eyed person, the one blue eyed person would look around and see only 999 brown eyed people plus the visitor. He would automatically know that he's the only blue-eyed dude in town and thus be compelled to commit suicide.

Day 2 - If there were only 2 blue eyed people on the island, the two blue eyed people would look around and see 1 other blue eyed person plus 998 brown eyed people. As long as no one committed suicide on the previous day, he would be able to deduce that he had blue eyes. (If no one committed suicide the prior day, then there must have been at least 1 other blue eyed person on that day. Noone has committed suicide, so he can also look around and see that there can be no more than 2 total blue eyed persons, including himself. He can only see one of them, so he must be the only other blue eyed person. Hence, he must die.)

Day 3 - If there were only 3 blue eyed people on the island, each of the three would look around and see 2 blue eyed people plus 997 brown eyed people. As long as no one committed suicide the prior 2 days, he can be assured that that he is one of 3 people that has blue eyes.

That process will continue reliably for the next 100 days. When the suicides stop, all the brown eyed people can be assured they have brown eyes.

Is that clearer?

No. If what you describe happens, it happens with or without the visitor. Which means that it would have happened years earlier and all the blue eyes would have been long gone, leaving only the brown eyes which now realize that everyone has brown eyes and they must also kill themselves. That means when the visitor arrives, there is no one alive on the island at all, and probably has not been for many years.

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ORIGINAL: DaveW

No not initially aware. But as the OP says:

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"highly logical" means that any conclusion that can logically deduced from the information and observations available to an islander, will automatically be known to that islander.

Simple observation would show those statistics.

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ORIGINAL: McFatty

He's addressing the entire tribe at that point... all 1,000 of them. He mentions that it's unusual to see another blue eyed person there, but everyone there already knows there are blue eyed people there, so the newcomer isn't alerting anyone to their own eye color.

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The punishment wouldn't be murder but a compulsion to commit ritual suicide. This ritual isn't even known to the newcomer, and since he is not part of their religion, non-deadly compulsion to commit suicide would likely be ineffective. Lastly, all the tribesmen are logical, and logic would alert them to the fact that the newcomer is unaware of that taboo, and that any of them could be in a similar situation were they to leave the island. I think the dude will be fine.