I have a fair coin and a weighted coin that lands heads 60% of the time. I grab a coin at random. I flip that coin 5 times and get heads 4 times and tails once.
What is the probability that the coin I grabbed was the weighted coin? (answer to 3 decimal places)
Binomial distribution, P(X) = nCx px qn-x
P(getting 4 heads and 1 tail with fair coin) = 5C4 x 0.54 x 0.5
= 0.15625
P(getting 4 heads and 1 tail with weighted coin) = 5C4 x 0.64 x 0.4
= 0.2592
Bayes' Theorem: P(A | B) = P(A & B) / P(B)
P(weighted coin | 4 heads and 1 tail) = P(selecting weighted coin and getting 4 heads and 1 tail) / [P(selecting weighted coin and getting 4 heads and 1 tail) + P(selecting weighted coin and getting 4 heads and 1 tail)]
= (0.5 x 0.2592) / [(0.5 x 0.2592) + (0.5 x 0.15625)]
= 0.6239
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