Question

(1 point) Rework problem 11 from section 3.4 of your text, involving the flipping of either a fair or weighted coin. Assume that the weighted coin yields a heads with probability 0.2. You select one of the two coins at random, and flip it 3 times, noting heads or tails with each flip.

What is the probability that the weighted coin was selected, given that all 3 flips turned up heads?

Answer 1

F: Event of selecting a fair coin

W: Event of selecting a weighted coin

You select one of the two coins at random

P(F) = Probability of selecting a fair coin = 1/2=0.5

H: Event of head in single toss;

P(H|F) =Probability of getting a heads given a fair coin is tossed 1/2 =0.5

P(W) = Probability of selecting a weighted coin = 1/2=0.5

P(H|W) = Probability of getting a heads given a weighted coin is tossed = 0.2

3H : Event of 3 flips turned heads ;( First flip head, second flip heads , third flip heads)

P(3H) = P(H)P(H)P(H)

P(3H|F)

= Probability of getting heads in all three flips when a fair coin is selected

= P(H|F) x P(H|F) x P(H|F) = 0.5 x 0.5 x 0.5 =0.125

P(3H|W)

= Probability of getting heads in all three flips when a weighted coin is selected

= P(H|W) x P(H|W) x P(H|W) = 0.2 x 0.2 x 0.2 = 0.008

probability that the weighted coin was selected, given that all 3 flips turned up heads = P(W|3H)

By Bayes theorem,

P(W)P(3H|W) = 0.5 x 0.008 = 0.004

P(F)P(3H|F) = 0.5 x 0.125 = 0.0625

probability that the weighted coin was selected, given that all 3 flips turned up heads = 0.060150376