GATE CS 2025 Set 1 Q.32: Probability and Statistics

A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, the probability $P(\text{head}) = 0.5$ and for a fake coin, $P(\text{head}) = 1$ . 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)

Your answer

Official answer and solution

Accepted answers: 0.49 to 0.51
From: Official GATE 2025 answer key

Solution

Apply Bayes' theorem to compute $P(\text{Fake} \mid HH)$ .

Let $F$ be the event of picking the fake coin and $R$ be the event of picking a regular coin:

Prior probabilities: $P(F) = \frac{1}{5} = 0.2$ and $P(R) = \frac{4}{5} = 0.8$ .
Likelihood of getting two heads ( $HH$ ): $P(HH \mid F) = 1^2 = 1$ and $P(HH \mid R) = (0.5)^2 = 0.25$ .

By Bayes' theorem:

P(F \mid HH) = \frac{P(F) P(HH \mid F)}{P(F) P(HH \mid F) + P(R) P(HH \mid R)} = \frac{0.2 \times 1}{0.2 \times 1 + 0.8 \times 0.25} = \frac{0.2}{0.2 + 0.2} = 0.5

Final answer

0.5

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 1 (CS1)
Question: 32 of 65
Section: Computer Science and Information Technology

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