8. Suppose the U.S. Mint puts out a report that says that 1 in 100 coins they mi

Question

8. Suppose the U.S. Mint puts out a report that says that 1 in 100 coins they mint are not "fair" meaning they do not have a 50% probability of landing on heads when flipped. You take a random coin and take as your null hypothesis Ho : P(H) = .5 You flip it 10 times and 9 times it lands on heads. This yields a p value of.02686 in a two-sided test. Assume the unfair coins have a 90% probability of landing on heads What is the probability that the coin is not a fair coin? (Hint: Use Bayes' Therem)

Explanation / Answer

Now, if we choose a coin then there are two possiblities, either it is a fair coin or an unfair coin, Prob of being fair is 0.99 and prob of it being unfair = 0.01 as claimed by US Mint. Now if is unfair then prob of succes is 0.9 and that of failure is 0.1 while for the othercase prob of success is 0.5 and prob of failure is also 0.5

Now using Bayes theorem P(A1|B) = P(A1)* P(H|A1) / (P(A1)* P(H|A1) + P(A2)* P(H|A2))

Let A1 = event that it is an unfair coin =0.01 and P(H|A1) = 0.9 and P(H|A2) = 0.5

Prob of coin being unfair = 0.01*0.9 / 0.01*0.9 + 0.99*0.5 = 0.02 as per Bayes theorem

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