GATE CS 2018 Q.25 (CS Q.15): Probability and Statistics

Two people, P and Q, decide to independently roll two identical dice, each with 6 faces, numbered 1 to 6. The person with the lower number wins. In case of a tie, they roll the dice repeatedly until there is no tie. Define a trial as a throw of the dice by P and Q. Assume that all 6 numbers on each dice are equi-probable and that all trials are independent. The probability (rounded to 3 decimal places) that one of them wins on the third trial is _____.

Your answer

Official answer and solution

Accepted answers: 0.021 to 0.024
From: Official GATE 2018 answer key

Solution

Calculate the probability of consecutive ties followed by a decisive outcome on the third trial.

In a single trial, P and Q independently roll a fair 6-sided die. The total number of outcomes is $6 \times 6 = 36$ .

A tie occurs when both roll the same number: $(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)$ (6 outcomes).

P(\text{tie}) = \frac{6}{36} = \frac{1}{6}

A decisive outcome (someone wins) occurs when the numbers are different:

P(\text{win}) = 1 - P(\text{tie}) = \frac{30}{36} = \frac{5}{6}

For the game to be won on the third trial, the first two trials must end in a tie, and the third trial must result in a win:

P = P(\text{tie}) \times P(\text{tie}) \times P(\text{win}) = \left(\frac{1}{6}\right) \times \left(\frac{1}{6}\right) \times \left(\frac{5}{6}\right) = \frac{5}{216}

Evaluating the fraction:

\frac{5}{216} \approx 0.023148\dots

Rounding to 3 decimal places gives 0.023.

Final answer

0.023

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2018 Computer Science and Information Technology (CS)
Question: 25 of 65, printed as CS Q.15
Section: Computer Science and Information Technology

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