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Derren Brown recently had a show in which he flipped ten heads in a row. He just flipped coins all day and waited for it to happen eventually. If I flip a fair coin, I should believe there's a 50% chance it will come up heads. If I flip it three times, I should believe there's a 12.5% chance it will come up heads three times. If I have eight goes at flipping it three times, it seems I should believe there's a 100% chance of flipping three heads. If that's right, what's wrong with being increasingly confident at the beginning of each set of flips that this will be the one in which I flip three heads? It's obviously a bad argument: every time I fip the coin, there's a 50% chance it will turn up heads. But how could it be rational for me to bet that during the course of a day of coin flipping I'll flip three heads eventually but not be rational for me to be increasingly confident that the next set of three flips will be of three heads as the day progresses?
Matthew

Yes, if I flip a fair coin 3 times I have a 1 in 2 3 (i.e. 1 in 8, i.e. 12.5%) chance of throwing three heads. How do we get that result? The rule is that if P and Q are independent events, then the chance of (P and Q) = chance of P x chance of Q. Likewise, if P, Q and R are independent events, then the chance of (P and Q and R) = chance of P x chance of Q x chance of R. If each of P, Q, R as a 1 in 2 chance, then the chance of (P and Q and R) is 1 in 2 3 . But, no, if I make 8 trials at throwing three heads I don't have a 100% chance of pulling it off. For the trials are independent events. And the chance of any one trial being successful is still 1 in 8, irrespective of what happened in the previous trials. Likewise, the chance of any one trial being un successful is 7 in 8, irrespective of what happened the previous trials. So the chance of eight trials being unsuccessful is (7/8) 8 , which is about 0.34. So the chance of getting three heads at least once in 8 trials is .66, i.e....

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