Let B be the event that the biased coin is selected, and F be the event that the fair coin is selected. Let H be the event that the coin shows heads.

We are given:

• P(H|B) = 1/4 (Probability of getting heads given the biased coin is selected)
• P(H|F) = 1/2 (Probability of getting heads given the fair coin is selected)
• P(B) = 1/2 (Probability of selecting the biased coin)
• P(F) = 1/2 (Probability of selecting the fair coin)

We want to find P(B|H), the probability that the coin is biased given that it showed heads. We can use Bayes' Theorem:

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

Substitute the given values:

P(B|H) = [(1/4) * (1/2)] / [(1/4) * (1/2) + (1/2) * (1/2)]

P(B|H) = (1/8) / (1/8 + 1/4)

P(B|H) = (1/8) / (1/8 + 2/8)

P(B|H) = (1/8) / (3/8)

P(B|H) = 1/3