Use Bayes' theorem to find the indicated probability.
-Pierre has three coins one of which is biased so that it turns up tails 60% of the time. If he randomly selects one of the coins, tosses it three times, and obtains three tails, what is the probability that this is the biased coin?
A) 0.216
B) 0.498
C) 0.464
D) 0.333

Question

Quizplus · Accepted Answer

Let A be the event that the selected coin is biased and B be the event that three tails are obtained. We want to find P(A|B).

By Bayes' theorem,

P(A|B) = P(B|A) * P(A)/P(B)

We know that P(A) = 1/3 since there are three coins and only one of them is biased. 
We also know that P(B|A) = (0.6)^3 = 0.216 since the biased coin turns up tails 60% of the time.

To find P(B), we need to use the law of total probability. Let C be the event that the selected coin is unbiased. Then,

P(B) = P(B|A) * P(A) + P(B|C) * P(C)
      = (0.6)^3 * (1/3) + (0.5)^3 * (2/3)
      = 0.216 + 0.125
      = 0.341

Therefore,

P(A|B) = P(B|A) * P(A)/P(B)
       = 0.216 * (1/3)/0.341
       = 0.464

So the answer is C.

Use Bayes' Theorem to Find the Indicated Probability