introduction to probability Joseph K. Blitzstein , Jessica Hwang A hat contains
ID: 3328364 • Letter: I
Question
introduction to probability Joseph K. Blitzstein , Jessica Hwang A hat contains 100 co ins, where at least 99 are fair, but there may be one that is doubleheaded (always landing Heads); if there is no such coin, then all 100 are fair. Let D be the event that there is such a coin, and suppose that P(D)-1/2. A coin is chosen uniformly at random. The chosen coin is flipped 7 times, and it lands Heads all 7 times. (a) Given this information, what is the probability that one of the coins is doubleheaded? (b) Given this information, what is the probability that the chosen coin is doubleheaded?Explanation / Answer
a) P(one of the coin is double headed) = 1/2
b) P(chosen coin is double headed) = P(heads 7 times in a row on double headed coin) / P(7 heads in a row on any coin)
P(heads 7 times in a row on double headed coin) = 0.5 x (1/100) x 1 = 0.005
P(7 heads in a row on any coin) = P(doubleheaded coin is there, it is chosen and 7 heads appear) + P(double headed coin is there, ordinary coin is chosen and 7 heads appear + P(double headed coin is not there and seven heads appear)
= (0.5x0.01x1 + 0.5x0.99x (1/2)7+ 0.5x1x(1/2)7
= 0.0128
P(chosen coin is double headed) = P(heads 7 times in a row on double headed coin) / P(7 heads in a row on any coin)
= 0.005/0.0128
= 0.3914
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