A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, P(head) = 0.

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GATE-CS is a competitive examination conducted in India. This question is from the 2025 GATE-CS Engineering Mathematics paper.

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This GATE-CS 2025 Engineering Mathematics question has a medium difficulty level and is marked as an important question.

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Question

A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, 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)

Accepted Answer

Answer: 0.50Concept — Bayes' Theorem: Bayes' theorem updates the probability of a hypothesis given new evidence:P(H|E) = P(E|H) × P(H) / P(E)Define events:F = chosen coin is fake, R = chosen coin is regular.HH = event of getting two heads.Prior probabilities:P(F) = 1/5, P(R) = 4/5Likelihoods (probability of HH given coin type):P(HH | F) = 1 × 1 = 1 (fake coin always gives heads)P(HH | R) = 0.5 × 0.5 = 0.25 (regular coin, both tosses independent)Total probability of HH (Law of Total Probability):P(HH) = P(HH|F)·P(F) + P(HH|R)·P(R)= 1 × (1/5) + 0.25 × (4/5)= 0.2 + 0.2 = 0.4Apply Bayes' Theorem:P(F | HH) = P(HH|F) × P(F) / P(HH) = (1 × 0.2) / 0.4 = 0.2 / 0.4 = 0.50Intuition: Although there are 4 regular and 1 fake coin, getting two heads is much more likely with the fake coin (probability 1 vs. 0.25). Observing HH strongly increases the likelihood of the fake coin, bringing its posterior probability from 1/5 = 0.20 up to 0.50.

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