Correct : 0.50

Explanation:
To find the probability that the chosen coin is fake given that it landed on heads twice, we apply Bayes' Theorem.

1. Define the Events:
• Let F be the event of choosing the Fake coin.
• Let R be the event of choosing a Regular coin.
• Let 2H be the event of getting two consecutive heads.

2. Determine Prior Probabilities:
There are 5 coins total (1 fake, 4 regular):
• P(F) = 1/5 = 0.2
• P(R) = 4/5 = 0.4 / 0.8

3. Determine Conditional Probabilities:
• For a fake coin, P(Head) = 1. Therefore, the probability of getting two heads is:
    P(2H | F) = 1 × 1 = 1
• For a regular coin, P(Head) = 0.5. Therefore, the probability of getting two heads is:
    P(2H | R) = 0.5 × 0.5 = 0.25

4. Calculate Total Probability of Getting Two Heads P(2H):
Using the Law of Total Probability:
    P(2H) = [P(F) × P(2H | F)] + [P(R) × P(2H | R)]
    P(2H) = (0.2 × 1) + (0.8 × 0.25)
    P(2H) = 0.2 + 0.2 = 0.4

5. Apply Bayes' Theorem to Find P(F | 2H):
    P(F | 2H) = [P(F) × P(2H | F)] / P(2H)
    P(F | 2H) = 0.2 / 0.4 = 0.5

Correction Note: Let us re-verify calculations based on standard fraction forms:
• Prior: P(F) = 1/5, P(R) = 4/5
• Conditional: P(2H|F) = 1, P(2H|R) = 1/4
• Total Probability: P(2H) = (1/5 × 1) + (4/5 × 1/4) = 1/5 + 1/5 = 2/5 = 0.4
• Final Ratio: (1/5) / (2/5) = 1/2 = 0.50

Conclusion:
The probability that the coin chosen is the fake coin is exactly 0.50.