If an unbiased dice is rolled thrice, then the probability of getting a greater number in the i^ th roll than the number obtained in the (i-1)^ th roll, i=2,3 , is equal to

Probability PYQ — JEE Main 2024 · 9 Apr · Shift 2 · Q39 with Solution

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Correct answer: A

Let the three outcomes of the die rolls be $x_1, x_2, x_3$ .

We need the probability that $x_2>x_1 \quad \text{and} \quad x_3>x_2.$

So the three numbers must be in strictly increasing order: $x_1<x_2<x_3.$
Total number of possible outcomes when a fair die is rolled thrice: $6^3=216.$
Now count favorable outcomes.

Since the numbers must be strictly increasing and each roll is from $\{1,2,3,4,5,6\}$ , we just need to choose any 3 distinct numbers from 6, and they will have exactly one increasing arrangement.

Number of ways: $\binom{6}{3}=20.$

Hence, favorable outcomes $=20$ .
Therefore, the required probability is $\frac{20}{216}=\frac{5}{54}.$
Checking options:
- A: $\frac{5}{54}$ ✅
- B: $\frac{2}{54}$
- C: $\frac{1}{54}$
- D: $\frac{3}{54}$

So the correct answer is A.