GATE CS 2014 Set 1 Q48 — Probability: Random Variables and Distributions

Let $d_1, d_2, d_3,$ and $d_4$ be the outcomes of the four fair six-sided dice, where $1 \le d_i \le 6$ for each $i \in \{1, 2, 3, 4\}$.

We want to find the number of integer solutions to the equation:

To simplify, we can define a new set of variables by measuring how far each die's outcome is from its maximum value of $6$. Let:

Since $1 \le d_i \le 6$, the new variables must satisfy $0 \le x_i \le 5$. Substituting $d_i = 6 - x_i$ into the sum equation gives:

Since the sum of these non-negative integers is $2$, the upper bound constraint $x_i \le 5$ is automatically satisfied for all $i$.

The number of non-negative integer solutions to $x_1 + x_2 + x_3 + x_4 = 2$ can be found using the stars and bars formula:

Alternatively, we can list the partitions of $2$ into $4$ non-negative integers:

• Case 1: One $2$ and three $0$s (e.g., $2, 0, 0, 0$). The number of permutations is $\dfrac{4!}{3!} = 4$.
• Case 2: Two $1$s and two $0$s (e.g., $1, 1, 0, 0$). The number of permutations is $\dfrac{4!}{2!2!} = 6$.

Total favorable outcomes = $4 + 6 = 10$.

Since the total number of outcomes when rolling four dice is $6^4 = 1296$, the probability of getting a sum of $22$ is:

Comparing this with $\dfrac{X}{1296}$, we get: