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I'm pretty sure your confusion about probability stems from you not understanding the mathematical concept of a "limit". Your sequence HHT has a 12.5% chance of

I'm pretty sure your confusion about probability stems from you not understanding the mathematical concept of a "limit". Your sequence HHT has a 12.5% chance of happening, but so does the sequence TTH and as the number of trials approaches infinity, you will get similar counts of the two because they have equivalent probabilities.

danbruc 10y ago
No, I will get similar counts with high probability but not surely. There are sequences of probability zero that do not converge. Think about it this way. Every coin toss in a sequence, finite or infinite, on its own can surly turn out to be heads, can't it? And all tosses are independent, aren't they? So why can't all tosses turn out heads? Unless you have a convincing argument why some tosses have to yield tails eventually, you have to deal with the fact that not all sequences of tosses converge to the expected probability.
- jgehrcke 10y ago
  > Every coin toss in a sequence, finite or infinite, on its own can surly turn out to be heads, can't it? I would agree if the sentence contained just the word "finite". The "or infinite" is where you are thinking too intuitively, and not mathematically correct anymore. The difference between "finite" and "infinite" is precisely the solution to this paradox in your mind. I hope to be able to point out that it is easy to correct for that with just a bit of structured (but maybe non-intuitive) thinking. One of the simplest and (I would argue most complete) definitions of "a probability of 0.5 for both test outcomes A and B" is that, given an infinitely large sample size, half of the samples show result A, while the other half shows result B. Think about this for a second, and I recommend to also use this opportunity to appreciate again that half of infinity is still infinity. This relates the mathematical concept of infinity to the definition of probability. With this definition, it probably feels like I so far just reworded your question. That may not be satisfying. So, I would like to encourage you to imagine that you have superpowers and can actually perform an infinite number of tests. You do that on a sunny day and observe that all test outcomes were the same: A. You call it a day and you can conclude (using the mathematical definition from above) in your diary of days-with-superpowers: "Today I have empirically determined that the test shows outcome A with a probability of 1". You might smile and add "Peter said that outcome A has a probability of 0.5, but I have proven him wrong". In other words: if you do an infinite number of tests, the normalized distribution of test results precisely is the probability distribution of test results. I think we have learned by now that the concepts of infinity and probability are deeply related and can, by definition, be used to explain each other. That might still not be satisfying. So, I would like to focus on the "finite" case for a bit. Imagine you don't have super powers anymore, but you're pretty resilient and motivated and you want to do the experiment to (in)validate Peter's claim: "The probability for both, outcome A and B, is 0.5 each!". After 1.333.337 tests you have seen 1.333.337 test results showing A. You're tired from all the testing and you complain (correctly!): "it is now really pretty unlikely that Peter is right! I am pretty damn sure that he is wrong! How long do I still need to do this to be absolutely sure?" -- and then a voice from the darknet reminds you: "for being absolutely sure, that is, for finding an answer that is correct with a probability of 1, you need to have super powers and make an infinite number of tests -- sorry dude, you can't do that, ever, because it's inconvenient, takes infinitely long and such -- so I need to disappoint you, you will never be sure, but maybe just enjoy your life as much as you can". Infinity usually does not allow for actual intuitive thinking. But there are a few really simple mathematical rules around infinity and convergence that make it actually pretty simple and again intuitive to deal with the concept.
  - danbruc 10y ago
    I appreciate your attempt but you did not convince me the slightest bit. Let's take the 1,333,337 coin tosses all heads. This result has no bearing on the probability of the coin at all. It may make you strongly doubt that the coin is indeed a fair coin but - and that is the point I am trying to get at - there is nothing that prevents a fair coin from coming up heads 1,333,337 times in a row. Whatever your experiment shows, it could always be a statistical fluke. And the infinite case does not changes much, at least not in a way obvious to me. Back with those super powers I toss the coin infinitely often and get heads 50 % of the time. That was fun, let me do that again tomorrow. Well, again heads 50 % of the time. This is the way it goes for a long time but then something strange happens, one day all tosses come up heads. The very next day everything is back to normal. What is now the probability of heads, we got two different answers for your way of defining he probability? And all it took was an extreme statistical outlier on single day.
    - kaoD 10y ago
      Not OP but I think you still don't grasp the concept of infinity. Terms like "the very next day" or any other segmentation don't apply.
    - yorwba 10y ago
      > What is now the probability of heads, we got two different answers for your way of defining he probability? And all it took was an extreme statistical outlier on single day. The point of probability is that performing an experiment an infinite number of times guarantees that every outcome happens with a proportion that exactly equals its probability (for a formalization of what that even means, look at measure theory). If you get different proportions on different days, you have different probabilities. That means, you weren't performing the same experiment.
  - srean 10y ago
    That's why I say you cant prove a probabilist/statistician wrong about probabilities of an outcome in any finite amount of time.... Its always good to be in such a business except for they cant be proven right either (in finite time)

danbruc 10y ago

No, I will get similar counts with high probability but not surely. There are sequences of probability zero that do not converge. Think about it this way. Every coin toss in a sequence, finite or infinite, on its own can surly turn out to be heads, can't it? And all tosses are independent, aren't they? So why can't all tosses turn out heads? Unless you have a convincing argument why some tosses have to yield tails eventually, you have to deal with the fact that not all sequences of tosses converge to the expected probability.

jgehrcke 10y ago
> Every coin toss in a sequence, finite or infinite, on its own can surly turn out to be heads, can't it? I would agree if the sentence contained just the word "finite". The "or infinite" is where you are thinking too intuitively, and not mathematically correct anymore. The difference between "finite" and "infinite" is precisely the solution to this paradox in your mind. I hope to be able to point out that it is easy to correct for that with just a bit of structured (but maybe non-intuitive) thinking. One of the simplest and (I would argue most complete) definitions of "a probability of 0.5 for both test outcomes A and B" is that, given an infinitely large sample size, half of the samples show result A, while the other half shows result B. Think about this for a second, and I recommend to also use this opportunity to appreciate again that half of infinity is still infinity. This relates the mathematical concept of infinity to the definition of probability. With this definition, it probably feels like I so far just reworded your question. That may not be satisfying. So, I would like to encourage you to imagine that you have superpowers and can actually perform an infinite number of tests. You do that on a sunny day and observe that all test outcomes were the same: A. You call it a day and you can conclude (using the mathematical definition from above) in your diary of days-with-superpowers: "Today I have empirically determined that the test shows outcome A with a probability of 1". You might smile and add "Peter said that outcome A has a probability of 0.5, but I have proven him wrong". In other words: if you do an infinite number of tests, the normalized distribution of test results precisely is the probability distribution of test results. I think we have learned by now that the concepts of infinity and probability are deeply related and can, by definition, be used to explain each other. That might still not be satisfying. So, I would like to focus on the "finite" case for a bit. Imagine you don't have super powers anymore, but you're pretty resilient and motivated and you want to do the experiment to (in)validate Peter's claim: "The probability for both, outcome A and B, is 0.5 each!". After 1.333.337 tests you have seen 1.333.337 test results showing A. You're tired from all the testing and you complain (correctly!): "it is now really pretty unlikely that Peter is right! I am pretty damn sure that he is wrong! How long do I still need to do this to be absolutely sure?" -- and then a voice from the darknet reminds you: "for being absolutely sure, that is, for finding an answer that is correct with a probability of 1, you need to have super powers and make an infinite number of tests -- sorry dude, you can't do that, ever, because it's inconvenient, takes infinitely long and such -- so I need to disappoint you, you will never be sure, but maybe just enjoy your life as much as you can". Infinity usually does not allow for actual intuitive thinking. But there are a few really simple mathematical rules around infinity and convergence that make it actually pretty simple and again intuitive to deal with the concept.
- danbruc 10y ago
  I appreciate your attempt but you did not convince me the slightest bit. Let's take the 1,333,337 coin tosses all heads. This result has no bearing on the probability of the coin at all. It may make you strongly doubt that the coin is indeed a fair coin but - and that is the point I am trying to get at - there is nothing that prevents a fair coin from coming up heads 1,333,337 times in a row. Whatever your experiment shows, it could always be a statistical fluke. And the infinite case does not changes much, at least not in a way obvious to me. Back with those super powers I toss the coin infinitely often and get heads 50 % of the time. That was fun, let me do that again tomorrow. Well, again heads 50 % of the time. This is the way it goes for a long time but then something strange happens, one day all tosses come up heads. The very next day everything is back to normal. What is now the probability of heads, we got two different answers for your way of defining he probability? And all it took was an extreme statistical outlier on single day.
  - kaoD 10y ago
    Not OP but I think you still don't grasp the concept of infinity. Terms like "the very next day" or any other segmentation don't apply.
  - yorwba 10y ago
    > What is now the probability of heads, we got two different answers for your way of defining he probability? And all it took was an extreme statistical outlier on single day. The point of probability is that performing an experiment an infinite number of times guarantees that every outcome happens with a proportion that exactly equals its probability (for a formalization of what that even means, look at measure theory). If you get different proportions on different days, you have different probabilities. That means, you weren't performing the same experiment.
- srean 10y ago
  That's why I say you cant prove a probabilist/statistician wrong about probabilities of an outcome in any finite amount of time.... Its always good to be in such a business except for they cant be proven right either (in finite time)