Comment on Infintiy Infintiy Infintiy Infintiy Infintiy Infintiy Infintiy Infintiy

• JackbyDev@programming.dev ⁨2⁩ ⁨weeks⁩ ago

  That’s assuming they’re typing truly randomly. Which is a fair assumption.

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• BluesF@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

  Not necessarily. Each monkey is independent, right? So if we think about the first letter, it’s either going to be, idk, A, the correct letter, or B, any wrong letter. Any monkey that types B is never going to get there. Now each money independently chooses between them. With each second monkey, the chances in aggregate get smaller and smaller than we only see B, but… It’s never a 0 chance that the monkey hits B. If there’s only two keys, it’s always 50/50. And it could through freak chance turn out that they all hit B… Forever. There is never a guarantee that you will get even a single correct letter… Even with infinite monkeys.

  I get that it seems like infinity has to include every possible outcome, because the limit of P(at least one monkey typing A) as the number of monkeys goes to infinity is 1… But a limit is not a value. The probability never reaches 1 even with infinite monkeys.

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  • lemonmelon@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

    The birthday problem fits into this somehow, but I can’t quite get there right now. Something like an inverse birthday problem to illustrate how, even though the probability of two monkeys typing the same letter rises quickly as more monkeys are added to the mix, and at a certain point (n+1, where n is “possible keystrokes”) it is inevitable that at least two monkeys will key identically, the inverse isn’t true.

    If you have 732 people in a room, there’s no guarantee that any one of them was born on August 12th.

    There’s another one that describes this better but it escapes me.

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  • NikkiDimes@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

    Infinite monkeys. Any probability greater than zero times infinity is infinity. You will see an infinite number of monkeys hitting A and an infinite number hitting B. If there were a finite number of monkeys, you would be correct, but that is not the case.

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    • BluesF@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

      No, that’s not how probability works. “Any probability times infinity is infinity” doesn’t even mean anything. Probabilities are between 0 and 1 so if for some reason you were to multiply an infinite number of them you would never end up with an “infinite” probability.

      I explained the infinity monkeys in another comment more clearly than I did above -here you go.

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      • NikkiDimes@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

        I could have worded that better. Any probability with a non-zero chance of occurring will occur an infinite number of times given an infinite sequence.

        To address the comment you linked, I understand what you’re saying, but you’re putting a lot of emphasis on something that might as well be impossible. In an infinite sequence of coin flips, the probability of any specific outcome - like all heads - is exactly zero. This doesn’t mean it’s strictly impossible in a logical sense; rather, in the language of probability, it’s so improbable that it effectively “never happens” within the probability space we’re working with. Theoretically, sure, you’re correct, but realistically speaking, it’s statistically irrelevant.

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• ChairmanMeow@programming.dev ⁨2⁩ ⁨weeks⁩ ago

  That’s not true. Infinite doesn’t mean “all”. There are an infinite amount of numbers between 0 and 1, but none of them are 2. There’s a high statistical probability, sure, but it’s not necessarily 100%.

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  • Dagrothus@reddthat.com ⁨2⁩ ⁨weeks⁩ ago

    It is necessarily 100%. That’s the whole idea behind infinity. There is a 0% chance of rolling a “2” because it’s outside the bounds of the question. Theres a 0% chance of the monkeys typing in chinese too.

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    • BluesF@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

      No, it isn’t, that’s a misunderstanding of how independent random variables behave. Even with an infinite number of trials, there is never a guarantee that any particular outcome will happen.

      Consider a coin flip, 50/50 chance of either getting heads or tails on each flip. Lets say we do an infinite number of flips, one by one, so that we end up with an infinite ordered set of outcomes, like so: {H, T, T, H, … }. Now, consider the probability of getting a particular arrangement of heads/tails in this infinite list, like the one I wrote before. You can’t calculate a probability for each arrangement - there are an infinite number - but it should be clear that each arrangement is equally likely, right? Because {H, …} is just as likely as {T, …}, same with {H, H, …} and {H, T, … } and so on and so on. In other words the probabilty of getting all heads on infinite coin flips is the same as the probability of getting any other combination.

      In the same way, the infinite monkeys are doing ‘coin flips’ involving more than 2 options. Lets just assume they have 26 keys, one for each letter, and assume they hit each of them with equal probability. In the same way, for an individual monkey the probability of going {a, a, a, a, a, a, …, a} is the same as the probability of the same sequence with hamlet somewhere (in a particular position that is - the probabilities are only equal when we consider exactly one arrangement). What might make it more intuitively clear is that even after an infinite number of trials you only have one sequence of letters (or set of sequences, with infinite monkeys). It’s clear that there are other possible sequences - like only the letter a - and these all have a non 0 chance of having arisen given a different infinite set of monkeys for a different infinite time period.

      It’s easy to be misled here! If we return to the coin flip example, the probability of flipping at least 1 head after infinite coin flips approaches 1. The limit of P(at least one H) as the number of flips approaches infinity is 1. But this is a limit! You never reach the limit, even considering infinite situations.

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      • Dagrothus@reddthat.com ⁨2⁩ ⁨weeks⁩ ago

        0.99999… repeating equals 1. Not close to 1. Equal to 1. The monkeys will necessarily type hamlet somewhere in the sequence. If your group of monkeys hasnt typed it yet, double the number of monkeys.

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