I have a bag with 3 coins in it. one of them is a fair coin, but the others are biased trick coins. when flipped, the three coins come up heads with probability 0.5, 0.3, 0.6 respectively. suppose that i pick one of these three coins entirely at random and flip it three times.
a. what is p(htt)? (i.e., it comes up heads on the first flip and tails on the last two flips.) assuming that the three flips, in order, are htt, what is the probability that the coin that i picked was the fair coin?
b.
Answers
Given : a bag with 3 coins in it.
One of them is a fair coin, but the others are biased trick coins. When flipped,
the three coins up heads with probability 0.5, 0.6 and 0.1 respectively. I pick one of these three coins uniformly at random and flip it three times.
To Find : (i) What is the probability that it comes up heads on the first flip and tails on the second and third flips?
(ii) Assuming that the three flips, in order, are HTT, what is the probability that the coin that I picked was the fair coin?
Solution:
Fair coin P(H) = 0.5 P(T) = 0.5
Biased coin1 P(H) = 0.6 P(T) = 0.4
Biased coin2 P(H) = 0.1 P(T) = 0.9
probability that it comes up heads on the first flip and tails on the second and third flips
= (1/3)(0.5)(0.5)(0.5) + (1/3)(0.4)(0.6)(0.6) + (1/3)(0.1)(0.9)(0.9)
= (1/3)( 0.125 + 0.144 + 0.081)
= (1/3) (0.35)
= 35/300
= 7/60
= 0.1667
probability that the coin that I picked was the fair coin =
(1/3)(0.125)/(1/3)0.35
= 125/350
= 25/70
= 5/14
= 0.357
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