4 ms·

From the problem description on Wikipedia. The director _randomly_ puts one prisoner's number in each closed drawer.

Retric 9y ago
Random does not mean independently chosen random number. Consider, if you pick a completely random location then two numbers could end up in the same box so one prisoner's number implies there is some order. Further, random does not mean even distribution. The logic is really based on a very specific kind of random distribution.
- pronoiac 9y ago
  If you shuffle a deck of cards, do you ever get a 2/5 of heart/diamonds?
  - Retric 9y ago
    One operation to shuffle a deck of cards is to cut it creating a random offset. So, I have seen someone just move the top few cards when asked to shuffle the deck. This makes a small offset more likely than you might assume. Remember the odds should be evenly spread across 52! which means in all of human history each deck should be unique, but in practice people have played with a default deck. (http://www.philly.com/philly/news/new_jersey/20120904_A_C__s_Golden_Nugget_in_legal_war_with__1_5_million_winners.html http://www.philly.com/philly/news/new_jersey/20120904_A_C__s...)
- kcanini 9y ago
  Colloquially, "random" does imply a uniform distribution.
  - Retric 9y ago
    Depends on context, I randomly tossed it does not imply all locations it might end up are equally likely. As it rolls after impact it's more likely to end up in a dip than on top of a hill etc. At best the direction is random but not necessarily the angle with respect to the ground. Basically, even distributions are very rare in the real world.
    - dmurray 9y ago
      Even if the warden's assignment of numbers to boxes is not random (perhaps he knows the suggested winning strategy and wants to foil it), the players can make it random by randomly permuting the numbers on the outside of the boxes. This guarantees their 31% success rate even against an adversarial warden.
      - Retric 9y ago
        Players can't communicate. So, this comes down to what that means, but your suggestion is reasonable and might depending on how the problem was carried out apply. aka, they can't all individually come up with the same random numbers without communicating. But, they could all independently come up with the same strategy if the all saw the same numbers on the outside of the boxes.
        danbruc 9y ago
        From the problem description on Wikipedia again. Before the first prisoner enters the room, the prisoners may discuss strategy—but may not communicate once the first prisoner enters to look in the drawers. So they certainly could all use a pre-agreed random permutation to compensate for any insufficient or evil [non-]randomness introduced by the director.
        Retric 9y ago
        I was not clear in my wording. They need to enter the room and know box one is this and thus I should treat that as box 22. But again without seeing the boxes they need a system to unambiguously order the boxes. That's not strictly a math problem as they are not say given exact x,y,z locations or precise measuring tools etc.
        danbruc 9y ago
        Or what if it is dark inside the room and they can not read the numbers in the drawers? Or what if there is no gravity in the room and they can not determine which drawers are at the top? Or what if the director used a script or number system none of the prisoners can read or understand? Or what if the numbers are only 100 nm tall inscribed with electron-beam lithography on small pieces of metal? Or what if there is no oxygen in the room? Or what if the room is 10,000 km long, wide, and high?

Random does not mean independently chosen random number. Consider, if you pick a completely random location then two numbers could end up in the same box so one prisoner's number implies there is some order. Further, random does not mean even distribution. The logic is really based on a very specific kind of random distribution.

pronoiac 9y ago
If you shuffle a deck of cards, do you ever get a 2/5 of heart/diamonds?
- Retric 9y ago
  One operation to shuffle a deck of cards is to cut it creating a random offset. So, I have seen someone just move the top few cards when asked to shuffle the deck. This makes a small offset more likely than you might assume. Remember the odds should be evenly spread across 52! which means in all of human history each deck should be unique, but in practice people have played with a default deck. (http://www.philly.com/philly/news/new_jersey/20120904_A_C__s_Golden_Nugget_in_legal_war_with__1_5_million_winners.html http://www.philly.com/philly/news/new_jersey/20120904_A_C__s...)
kcanini 9y ago
Colloquially, "random" does imply a uniform distribution.
- Retric 9y ago
  Depends on context, I randomly tossed it does not imply all locations it might end up are equally likely. As it rolls after impact it's more likely to end up in a dip than on top of a hill etc. At best the direction is random but not necessarily the angle with respect to the ground. Basically, even distributions are very rare in the real world.
  - dmurray 9y ago
    Even if the warden's assignment of numbers to boxes is not random (perhaps he knows the suggested winning strategy and wants to foil it), the players can make it random by randomly permuting the numbers on the outside of the boxes. This guarantees their 31% success rate even against an adversarial warden.
    - Retric 9y ago
      Players can't communicate. So, this comes down to what that means, but your suggestion is reasonable and might depending on how the problem was carried out apply. aka, they can't all individually come up with the same random numbers without communicating. But, they could all independently come up with the same strategy if the all saw the same numbers on the outside of the boxes.
      - danbruc 9y ago
        From the problem description on Wikipedia again. Before the first prisoner enters the room, the prisoners may discuss strategy—but may not communicate once the first prisoner enters to look in the drawers. So they certainly could all use a pre-agreed random permutation to compensate for any insufficient or evil [non-]randomness introduced by the director.
        Retric 9y ago
        I was not clear in my wording. They need to enter the room and know box one is this and thus I should treat that as box 22. But again without seeing the boxes they need a system to unambiguously order the boxes. That's not strictly a math problem as they are not say given exact x,y,z locations or precise measuring tools etc.
        danbruc 9y ago
        Or what if it is dark inside the room and they can not read the numbers in the drawers? Or what if there is no gravity in the room and they can not determine which drawers are at the top? Or what if the director used a script or number system none of the prisoners can read or understand? Or what if the numbers are only 100 nm tall inscribed with electron-beam lithography on small pieces of metal? Or what if there is no oxygen in the room? Or what if the room is 10,000 km long, wide, and high?