Consider 10 independent flips of a fair coin. These are two possible outcomes: O

Question

Consider 10 independent flips of a fair coin. These are two possible outcomes:
Outcome 1: HTHTHTHTHT
Outcome 2: HTHHTHTTHT

(a) Which of the two patterns above is more believable as an outcome?

(b) Compute the probability of each, is it consistent with your answer for part (a)?

(c) Consider the possibility that the coin flips are not independent with P(T|H) = P(H|T) = p > 1/2, and that P(independent) = P(dependent) = 1/2 (zero knowledge). Show how we can resolve the inconsistency by using a Bayesian approach.

Explanation / Answer

a) Since there is equal probability of H and T then P(H)= P(T)= 1/2= 0.5 So they both have the same probability as it does not matter whether it's H or T or which order they're in. b) O1= HTHTHTHTHT P(O1)= 0.5*0.5*0.5*0.5*0.5*0.5*0.5*0.5*0.5*0.5= (0.5)^10 O2= HTHHTHTTHT P(O2)= 0.5*0.5*0.5*0.5*0.5*0.5*0.5*0.5*0.5*0.5= (0.5)^10 Therefore, they DO have the same probability and the answer is consistent with the answer to part(a).

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