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I understand that each toss has a 50% chance if it is a fair coin, but I have hard time grasping the law of great numbers and I would like to know how likely it is that I get 20 heads in a row in such a large number of tosses.

Btw, is there really any difference if the coin IS NOT fair? Lets say that I only have a 47.5% chance of winning, does that increase the probability of getting 20 heads in a row?

Regards,

Dionysious
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    If the coin is not fair then there is indeed a difference. E.g. let it be a coin having heads on both sides. – drhab Jul 13 '16 at 11:27
  • Also have a look at this question – drhab Jul 13 '16 at 11:39
  • One can use combinatorics when there are two outcomes and its fair; generally the probability measure and the proportion of possible case line up; every sequence of outcomes being equi-probable; just calculate the number of possible sequences; and the proportion that have 20 heads in a row (it also depends on what you mean by row) if you mean the probability that any arbitrary such sequence, has one such row then its bound to be higher, ie that eventually there will be such a row, , but if you mean in general; that for any arbitrary row, of 20 tosses, its different; – William Balthes May 07 '17 at 05:17
  • i presume you mean the former, as the 100 million tosse would be, irrelevant to the answer then – William Balthes May 07 '17 at 05:18

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Hint for your first question:

Write $0$ if tais and $1$ if heads. Then the results of the 100-million coin tosses can be represented as a bitstring.

The probability of all possible 100-million digit bitstrings appearing are equal, since each coin flip is fair.

So you only have to count the number of 100-million digit bitstrings with at least one 20 digit subsequence of purely $1$s.

Edit: It turns out that you can use this recursive formula to calculate the probability of a run of K events in a row at least once in a series of N trials. In the final step of my hint, one may even have to resort to this formula in the end.
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Yiyuan Lee
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  • I wonder how to do that using pencil & paper. – Pavel Vergeev Jul 13 '16 at 11:20
  • Is there any difference if the subject chooses always heads or if he keeps alternating from heads to tails at each toss? Although intuitively I think that it shouldn't really matter, is there any counterintuitive gain from alternating heads to tails at each toss? – Dionysious Jul 14 '16 at 05:26