jee-advanced 2022 Q16

jee-advanced · India · paper1 3 marks Probability Definitions Finite Equally-Likely Probability Computation
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 $i ^ { \text {th } }$ round.
List-I (I) Probability of $\left( X _ { 2 } \geq Y _ { 2 } \right)$ is (II) Probability of $\left( X _ { 2 } > Y _ { 2 } \right)$ is (III) Probability of $\left( X _ { 3 } = Y _ { 3 } \right)$ is (IV) Probability of $\left( X _ { 3 } > Y _ { 3 } \right)$ is
List-II (P) $\frac { 3 } { 8 }$ (Q) $\frac { 11 } { 16 }$ (R) $\frac { 5 } { 16 }$ (S) $\frac { 355 } { 864 }$ (T) $\frac { 77 } { 432 }$
The correct option is:
(A) (I) → (Q); (II) → (R); (III) → (T); (IV) → (S)
(B) (I) → (Q); (II) → (R); (III) → (T); (IV) → (T)
(C) (I) → (P); (II) → (R); (III) → (Q); (IV) → (S)
(D) (I) → (P); (II) → (R); (III) → (Q); (IV) → (T)
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 $i ^ { \text {th } }$ round.

\textbf{List-I}\\
(I) Probability of $\left( X _ { 2 } \geq Y _ { 2 } \right)$ is\\
(II) Probability of $\left( X _ { 2 } > Y _ { 2 } \right)$ is\\
(III) Probability of $\left( X _ { 3 } = Y _ { 3 } \right)$ is\\
(IV) Probability of $\left( X _ { 3 } > Y _ { 3 } \right)$ is

\textbf{List-II}\\
(P) $\frac { 3 } { 8 }$\\
(Q) $\frac { 11 } { 16 }$\\
(R) $\frac { 5 } { 16 }$\\
(S) $\frac { 355 } { 864 }$\\
(T) $\frac { 77 } { 432 }$

The correct option is:\\
(A) (I) → (Q); (II) → (R); (III) → (T); (IV) → (S)\\
(B) (I) → (Q); (II) → (R); (III) → (T); (IV) → (T)\\
(C) (I) → (P); (II) → (R); (III) → (Q); (IV) → (S)\\
(D) (I) → (P); (II) → (R); (III) → (Q); (IV) → (T)