One of three coins is chosen at random: a fair coin, a two-headed coin, and a coin with . The chosen coin is tossed twice and shows heads both times. The probability that it is the two-headed coin is
Answer and explanation
Correct answer: A.
- Fair coin : .
- Two-headed coin : .
- Biased coin : .
- Two-headed coin : .
- Biased coin : .
- Step 1: Let the three equally likely coin choices have prior probability . Compute the probability of obtaining two heads () for each coin:
- Step 2: Apply the law of total probability: .
- Step 3: Use Bayes' theorem to find the posterior probability that the coin is two-headed: .
Answer
.
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