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>That the relative frequency converges to 50 % is obviously not true. You would have to be exceptionally lucky, but you could get heads, heads, tails repeated f

>That the relative frequency converges to 50 % is obviously not true. You would have to be exceptionally lucky, but you could get heads, heads, tails repeated for ever This would violate the law of large numbers. You may end up with the sequence HHT 10million times, but the chances that you continue to get that sequence for another 10million times is all but zero, and then gets even smaller as you add another 10million trials. As the number of trials approaches infinity, you will arrive at 50%.

danbruc 10y ago
No, the law of large numbers does not assert that all sequences of outcomes converge, only that this happens almost surely. Or look at it the other way round, what mechanism would prevent heads, heads, tails repeated forever? I can certainly get heads, heads, tails on the first three tosses. After that I start over, three more tosses all independent of what just happened, again a 12.5 % chance for heads, heads, tails. Why could this not continue forever?
- hellogoodbyeeee 10y ago
  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

No, the law of large numbers does not assert that all sequences of outcomes converge, only that this happens almost surely. Or look at it the other way round, what mechanism would prevent heads, heads, tails repeated forever? I can certainly get heads, heads, tails on the first three tosses. After that I start over, three more tosses all independent of what just happened, again a 12.5 % chance for heads, heads, tails. Why could this not continue forever?

hellogoodbyeeee 10y ago
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.