You are asked by a friend to show him the coins you have in your pocket or purse and you find two ten cent coins. He selects one of them, and then proceeds to toss it ten times, obtaining eight heads and two tails in some order. He then offers to bet on the results of tossing the coin again. Feeling that the data is too restricted to allow you to estimate odds sensibly, you toss the other coin one thousand times (rather quickly) and keep count of the results. You get 503 Heads and 497 Tails, but you note that the variation if you partition them into groups of ten is considerable and accords well with the predictions of a binomial model with p(H) = 0.5. Present a careful argument to determine what odds you would offer your friend in a bet for a dollar on the next throw of his coin coming heads. Repeat for the case where the bet is one hundred dollars. Would it make a difference if the friend was called Persi? If you were allowed to be the one to toss the coin? Could the order of the results make a difference, for example if the two tails came first? Last? What about the possibility that the number of throws was chosen not because ten is a nice round number but because your friend threw two tails at the end because he was getting tired and thought that two bad throws meant it was time to stop?
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You work to one bit precision on the unit interval and you have three data points,
With the assumptions of the last question, how many points are needed in order to justify the use of the appropriate binomial model as a compression device?
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