An experiment consists of randomly selecting one of three coins, tossing it, and observing the outcome - heads or tails. The first coin is a two-headed coin, the second is a biased coin such that P(H) = , and the third is a fair coin. If the coin selected shows heads, what is the probability that this coin is the fair coin?

Question

An experiment consists of randomly selecting one of three coins, tossing it, and observing the outcome - heads or tails. The first coin is a two-headed coin, the second is a biased coin such that P(H) =  , and the third is a fair coin. If the coin selected shows heads, what is the probability that this coin is the fair coin?

Quizplus · Accepted Answer

We can use Bayes' theorem to find the probability that the selected coin is the fair coin given that a head was observed. Let F, B, and T represent the events of selecting the fair coin, biased coin, and two-headed coin, respectively. We want to find P(F|H), the probability that the fair coin was selected given that a head was observed. By Bayes' theorem:

P(F|H) = P(H|F) * P(F) / [P(H|F)*P(F) + P(H|B)*P(B) + P(H|T)*P(T)]

We are given that P(H|F) = 1/2 (since the fair coin is a fair coin), P(H|B) = 11eab448_e7ec_4f46_acdb_515f78e6174a_TB6027_11 (since this is the probability of heads for the biased coin), P(H|T) = 1 (since the two-headed coin always shows heads), and P(F) = P(B) = P(T) = 1/3 (since each coin is equally likely to be selected). Substituting these values, we get:

P(F|H) = (1/2) * (1/3) / [(1/2)*(1/3) + 11eab448_e7ec_4f46_acdb_515f78e6174a_TB6027_11*(1/3) + 1*(1/3)]

Simplifying, we get:

P(F|H) = 11eab448_e7ec_4f48_acdb_c716008cfa07_TB6027_11

Therefore, the best choice is B.

An Experiment Consists of Randomly Selecting One of Three Coins