GATE 2026 CS (CS2) – Question 63
Suppose an unbiased coin is tossed 6 times. Each toss is independent. Let $E_1$ be the event that among the second, fourth, and sixth tosses, there are at least two heads. Let $E_2$ be the event that among the first, second, third, and fifth tosses, there are equal numbers of heads and tails. The conditional probability $P(E_1 \mid E_2)$ is equal to __________. (rounded off to one decimal place)
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Correct answer: 0.5
Explanation
Event $E_2$ means that among tosses 1, 2, 3, and 5 there are exactly two heads and two tails. There are $\binom{4}{2}=6$ such outcomes, all equally likely. Given $E_2$, toss 2 is a head with probability $\frac{1}{2}$. If toss 2 is a head, then among tosses 4 and 6 we need at least one head, which has probability $\frac{3}{4}$. If toss 2 is a tail, then tosses 4 and 6 must both be heads, which has probability $\frac{1}{4}$. Therefore, $$P(E_1\mid E_2)=\frac{1}{2}\cdot\frac{3}{4}+\frac{1}{2}\cdot\frac{1}{4}=\frac{1}{2}.$$ Hence the answer is 0.5.