A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, the probability \(P(head) = 0.5\) and for a fake coin, \(P(head) = 1\). You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is _______. (rounded off to two decimal places)
Answer
0.49 to 0.51
Explanation
Use Bayes’ theorem, with \(F\) = the fake coin is chosen and \(HH\) = two heads.
\(P(F) = \frac{1}{5}\) and \(P(HH \mid F) = 1\); a regular coin is chosen with probability \(\frac{4}{5}\) and gives two heads with probability \(0.5 \times 0.5 = \frac{1}{4}\).
\(P(HH) = \frac{1}{5} \cdot 1 + \frac{4}{5} \cdot \frac{1}{4} = \frac{2}{5}\), so \(P(F \mid HH) = \dfrac{1/5}{2/5} = \frac{1}{2} = 0.50\).
Answer: 0.50