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)

Let $R$ denote the event of choosing a regular coin and $F$ denote the event of choosing the fake coin.

The prior probabilities are:

Let $E$ be the event of getting two consecutive heads ($HH$).

The conditional probabilities are:

Using Bayes' theorem, the posterior probability that the chosen coin is fake given two heads is:

Substitute the known values:

Rounded to two decimal places, the answer is 0.50 (acceptable range: $0.49$ to $0.51$).