GATE 2025 EC Q.58: solution and answer

GATE 2025 · ECQ.58 · 2 marks · NAT

Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained.

The expectation of $X$ is ______ (rounded off to two decimal places).

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

The expectation of the sum of two fair dice is $E [X] = 7.00$ .

Official answer: 6.95 to 7.05Source: Official answer key, GATE 2025 EC (IIT Roorkee)

Write $X = X_{1} + X_{2}$ , where $X_{1}$ and $X_{2}$ are the faces shown by the first and second die.
Each face of a fair die has probability $\frac{1}{6}$ , so $E [X_{1}] = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5$ . The same holds for $X_{2}$ .
By linearity of expectation, which holds whether or not the dice are independent, $E [X] = E [X_{1}] + E [X_{2}] = 3.5 + 3.5 = 7$ .
Check with the distribution of the sum. Of the $36$ equally likely outcomes, $6 - ∣ k - 7∣$ give the sum $k$ , so $P (X = k) = \frac{6 - ∣ k - 7∣}{36}$ for $k = 2, \dots, 12$ . These probabilities add to $\frac{1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1}{36} = 1$ . The distribution is symmetric about $k = 7$ , so its mean is $7$ .
Rounded off to two decimal places: $E [X] = 7.00$ .

KEY IDEA

Linearity of expectation: the mean of a sum is the sum of the means, $E [X_{1} + X_{2}] = E [X_{1}] + E [X_{2}]$ , so there is no need to build the distribution of the sum. One fair die has mean $3.5$ , so two dice have mean $7$ .

WATCH OUT FOR

Averaging only the 11 possible sums, $2$ to $12$ , as if each were equally likely. The mean happens to come out as $7$ here, but that is luck from symmetry: the sums are not equally likely ( $P (X = 7) = \frac{6}{36}$ while $P (X = 2) = \frac{1}{36}$ ), and the same method fails for any non-symmetric case. A second mistake is thinking linearity needs the dice to be independent. It does not; independence matters for the variance, not the mean.

From the GATE 2025 EC paper. Every question of this paper · Download the full paper · All 33 Probability & statistics questions, with notes

GATE 2025 EC Q.58: solution and answer

Answer and solution

More Probability & statistics questions