GATE CS 1996 Q1.5 — Probability: Random Variables and Distributions

When two fair dice are thrown simultaneously, the total number of possible outcomes in the sample space $S$ is:

We want to find the probability of the event $E$ that at least one of the dice shows a $6$.

Method 1: Using the Complement

The complement of "at least one $6$" is "no $6$ on either die".

• The probability of not getting a $6$ on the first die is $\frac{5}{6}$.
• The probability of not getting a $6$ on the second die is $\frac{5}{6}$.

Since the two rolls are independent, the probability of getting no $6$ on both dice is:

Therefore, the probability of getting at least one $6$ is:

Method 2: Direct Counting

The favorable outcomes where at least one die shows a $6$ are:

Counting these outcomes gives exactly $11$ favorable outcomes.

Thus, the correct option is D.