A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin…

Answer: 0.50 (probability that the chosen coin is the fake coin, rounded to two decimal places)

Explanation using Bayes' theorem:

• Prior probabilities: choosing the fake coin = 1/5; choosing a regular coin = 4/5.

• Likelihoods of observing two heads: if the coin is fake, probability = 1; if the coin is regular, probability = 0.5 × 0.5 = 0.25.

• Apply Bayes' theorem: posterior = (prior of fake × likelihood of two heads given fake) ÷ (total probability of two heads).

• Compute values: numerator = (1/5) × 1 = 0.2. Denominator = (1/5) × 1 + (4/5) × 0.25 = 0.2 + 0.2 = 0.4. So posterior = 0.2 ÷ 0.4 = 0.5.

• Rounded to two decimal places, the probability is 0.50.

Intuition: observing two heads makes the fake coin much more plausible because regular coins are relatively unlikely to produce two heads in a row.