JEE Math Practice Question

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:

1. A.(I) $\rightarrow$ (Q); (II) $\rightarrow$ (R); (III) $\rightarrow$ (T); (IV) $\rightarrow$ (S)
2. B.(I) $\rightarrow$ (Q); (II) $\rightarrow$ (R); (III) $\rightarrow$ (T); (IV) $\rightarrow$ (T)
3. C.(I) $\rightarrow$ (P); (II) $\rightarrow$ (R); (III) $\rightarrow$ (Q); (IV) $\rightarrow$ (S)
4. D.(I) $\rightarrow$ (P); (II) $\rightarrow$ (R); (III) $\rightarrow$ (Q); (IV) $\rightarrow$ (T)