(Mathematical statistics) Hi, would you mind answering the question? Thank you s

Question

(Mathematical statistics)

Hi, would you mind answering the question?

Thank you so much.

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.

Explanation / Answer

Solution:

Probability of (HTT) when fair coin (0.5) is flipped first = 0.5 * 0.3 * 0.6

p1 = 0.09

Probability of (HTT) when biased coin (0.3) is flipped first = 0.3 * 0.5 * (1 - 0.6)

p2 = 0.06

Probability of (HTT) when biased coin (0.6) is flipped first = 0.4* 0.5 * (1 - 0.3)

p3 = 0.14

total probability of getting HTT from the three coins P = p1 + p2 + p3

= 0.236

Assuming that the three flips in order HTT, Probability of picking a fair coin =

P (picking fair coin | HHT) = p1 / (p1 + p2 + p3)

= 0.09 / 0.236

= 0.381

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