JEE Advanced · Mathematics · 32. Probability

Two players, $P_{1}$ and $P_{2}$ , play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $x$ and $y$ denote the readings on the die rolled by $P_{1}$ and $P_{2}$ , respectively. If $x > y$ , then $P_{1}$ scores $5$ points and $P_{2}$ scores $0$ point. If $x = y$ , then each player scores $2$ points. If $x < y$ , then $P_{1}$ scores $0$ point and $P_{2}$ scores $5$ points. Let $X_{i}$ and $Y_{i}$ be the total scores of $P_{1}$ and $P_{2}$ , respectively, after playing the $i^{th}$ round.

List- $I$ List- $II$

$(I)$ Probability of $(X_{2} \geq Y_{2})$ is $(P)$ $\frac{3}{8}$

$(II)$ Probability of $(X_{2} > Y_{2})$ is $(Q)$ $\frac{11}{16}$

$(III)$ Probability of $(X_{3} = Y_{3})$ is $(R)$ $\frac{5}{16}$

$(IV)$ Probability of $(X_{3} > Y_{3})$ is $(S)$ $\frac{355}{864}$

$(T)$ $\frac{77}{432}$

The correct option is:

A $(I) \to (Q); (II) \to (R); (III) \to (T); (IV) \to (S)$
B $(I) \to (Q); (II) \to (R); (III) \to (T); (IV) \to (T)$
C $(I) \to (P); (II) \to (R); (III) \to (Q); (IV) \to (S)$
D $(I) \to (P); (II) \to (R); (III) \to (Q); (IV) \to (T)$

Verified Solution

Answer & Solution

Correct Answer

(A) $(I) \to (Q); (II) \to (R); (III) \to (T); (IV) \to (S)$

Step-by-step Solution

Detailed explanation

Given, $P$ (draw in $1$ round) $= \frac{6}{36} = \frac{1}{6}$

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Explore more questions on app

From JEE Advanced

Explore more questions on app

	List- $I$		List- $II$
$(I)$	Probability of $(X_{2} \geq Y_{2})$ is	$(P)$	$\frac{3}{8}$
$(II)$	Probability of $(X_{2} > Y_{2})$ is	$(Q)$	$\frac{11}{16}$
$(III)$	Probability of $(X_{3} = Y_{3})$ is	$(R)$	$\frac{5}{16}$
$(IV)$	Probability of $(X_{3} > Y_{3})$ is	$(S)$	$\frac{355}{864}$
		$(T)$	$\frac{77}{432}$