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An urn contains two types of coins. Coin type A and coin type B. When a type A coin is flipped, it comes up heads with probability 1/4, whereas when a type B coin is flipped, it comes up heads with probability 3/4. A coin is randomly chosen from the urn and flipped. Given that the flip landed on heads, the probability that it was a type A coin is:

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This question has been regraded.

An urn contains two types of coins. Coin type A and coin type B. When a type A coin is flipped, it comes up heads with probability 1/4, whereas when a type B coin is flipped, it comes up heads with probability 3/4. A coin is randomly chosen from the urn and flipped. Given that the flip landed on heads, the probability that it was a type A coin is:

BlackTom AI · Accepted Answer

The problem describes an urn with two coin types, A and B, chosen at random. When type A is flipped, P(H|A) = 1/4; when type B is flipped, P(H|B) = 3/4. Since a coin is randomly chosen from the urn, we assume P(A) = P(B) = 1/2.

Step 1: Compute the overall probability of heads, P(H). Using the law of total probability: P(H) = P(H|A)P(A) + P(H|B)P(B) = (1/4)(1/2) + (3/4)(1/2) = 1/8 + 3/8 = 4/8 = 1/2.

Step 2: Apply Bayes' rule to find P(A|H): P(A|H) = [P(H|A)P(A)] / P(H) = [(1/4)(1/2)] / (1/2) = (1/8) / (1/2) = (1/8) × 2 = 1/4.

Even though the answer options are not provided in the prompt, this calculation shows how P(A|H) is determined given the stated probabilities and the equal prior for selecting coin types. The key ideas are the law of total probability to obtain P(H) and Bayes’ theorem to update the probability of having chosen type A after observing heads.

If you were listing options, you would expect the correct choice to be the one equal to 1/4, since it directly follows from the Bayes calculation above. The reasoning hinges on the equal prior for A and B and the different head probabilities for the two coin types.

STAT2011 (ND) Early Feedback Task (EFT)

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