GATE CS 2026 Set 2 Q.63: Probability and Statistics

Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all previous coin tosses. Let $E_1$ be the event that among the second, fourth, and sixth coin tosses, there are at least two heads. Let $E_2$ be the event that among the first, second, third, and fifth coin tosses, there are equal number of heads and tails.

The conditional probability $P(E_1 \mid E_2)$ is equal to ____________. (rounded off to one decimal place)

Your answer

Official answer and solution

Official answer: 0.5
From: Official GATE 2026 answer key

Solution

Use the symmetry of fair coin tosses under complementing Heads and Tails ( $H \leftrightarrow T$ ).

Notice that under the bijection that flips every coin toss ( $H \leftrightarrow T$ ):

Event $E_2$ (having exactly 2 Heads and 2 Tails among tosses $\{1, 2, 3, 5\}$ ) is invariant ( $E_2$ holds for an outcome $\omega$ if and only if $E_2$ holds for $\overline{\omega}$ ).
Event $E_1$ (at least 2 Heads among the 3 tosses $\{2, 4, 6\}$ ) is mapped to its exact complement $\overline{E_1}$ (at most 1 Head, i.e., at least 2 Tails among $\{2, 4, 6\}$ ).

Since all outcomes are equally likely and $E_2$ is invariant under $H \leftrightarrow T$ , exactly half of the outcomes in $E_2$ belong to $E_1$ :

P(E_1 \mid E_2) = \frac{|E_1 \cap E_2|}{|E_2|} = \frac{1}{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 2026 Computer Science and Information Technology - Session 2 (CS2)
Question: 63 of 65
Section: Computer Science and Information Technology

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