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Official PYQ · 2025Set 1 · Q22Probability: Random Variables and DistributionsNAT1 mark

GATE CS 2025 Set 1 Q22 — Probability: Random Variables and Distributions

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A box contains 55 coins: 44 regular coins and 11 fake coin. When a regular coin is tossed, the probability P(P( head )=0.5)=0.5 and for a fake coin, P(P( head )=1)=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)
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Answer: 0.49:0.51

Let FF be the event that the chosen coin is fake, and RR be the event that it is regular. The prior probabilities are:

P(F)=15=0.2P(R)=45=0.8\begin{aligned}P(F) &= \frac{1}{5} = 0.2 \\ P(R) &= \frac{4}{5} = 0.8\end{aligned}

Let H2H_2 be the event of getting two heads in two tosses. The conditional probabilities are:

P(H2∣F)=12=1P(H2∣R)=(0.5)2=0.25\begin{aligned}P(H_2 \mid F) &= 1^2 = 1 \\ P(H_2 \mid R) &= (0.5)^2 = 0.25\end{aligned}

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

P(F∣H2)=P(H2∣F)P(F)P(H2∣F)P(F)+P(H2∣R)P(R)=1×0.2(1×0.2)+(0.25×0.8)=0.20.2+0.2=0.20.4=0.5\begin{aligned}P(F \mid H_2) &= \frac{P(H_2 \mid F) P(F)}{P(H_2 \mid F) P(F) + P(H_2 \mid R) P(R)} \\ &= \frac{1 \times 0.2}{(1 \times 0.2) + (0.25 \times 0.8)} \\ &= \frac{0.2}{0.2 + 0.2} = \frac{0.2}{0.4} = 0.5\end{aligned}

Thus, the probability that the chosen coin is fake is 0.5 (or 0.50).

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