1
JEE Advanced 2022 Paper 1 Online
+3
-1

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

The correct option is:

A
(I) $$\rightarrow$$ (Q); (II) $$\rightarrow$$ (R); (III) $$\rightarrow$$ (T); (IV) $$\rightarrow(S)$$
B
(I) $$\rightarrow$$ (Q); (II) $$\rightarrow$$ (R); (III) $$\rightarrow$$ (T); (IV) $$\rightarrow$$ (T)
C
(I) $$\rightarrow$$ (P); (II) $$\rightarrow$$ (R); (III) $$\rightarrow(\mathrm{Q}) ;(\mathrm{IV}) \rightarrow(\mathrm{S})$$
D
(I) $$\rightarrow$$ (P); (II) $$\rightarrow$$ (R); (III) $$\rightarrow$$ (Q); (IV) $$\rightarrow$$ (T)
2
JEE Advanced 2021 Paper 1 Online
+3
-1
Consider three sets E1 = {1, 2, 3}, F1 = {1, 3, 4} and G1 = {2, 3, 4, 5}. Two elements are chosen at random, without replacement, from the set E1, and let S1 denote the set of these chosen elements. Let E2 = E1 $$-$$ S1 and F2 = F1 $$\cup$$ S1. Now two elements are chosen at random, without replacement, from the set F2 and let S2 denote the set of these chosen elements.

Let G2 = G1 $$\cup$$ S2. Finally, two elements are chosen at random, without replacement, from the set G2 and let S3 denote the set of these chosen elements.

Let E3 = E2 $$\cup$$ S3. Given that E1 = E3, let p be the conditional probability of the event S1 = {1, 2}. Then the value of p is
A
$${1 \over 5}$$
B
$${3 \over 5}$$
C
$${1 \over 2}$$
D
$${2 \over 5}$$
3
JEE Advanced 2020 Paper 1 Offline
+3
-1
Let C1 and C2 be two biased coins such that the probabilities of getting head in a single toss are $${{2 \over 3}}$$ and $${{1 \over 3}}$$, respectively. Suppose $$\alpha$$ is the number of heads that appear when C1 is tossed twice, independently, and suppose $$\beta$$ is the number of heads that appear when C2 is tossed twice, independently. Then the probability that the roots of the quadratic polynomial x2 $$-$$ ax + $$\beta$$ are real and equal, is
A
$${{40} \over {81}}$$
B
$${{20} \over {81}}$$
C
$${{1} \over {2}}$$
D
$${{1} \over {4}}$$
4
JEE Advanced 2018 Paper 1 Offline
+3
-1
There are five students S1, S2, S3, S4 and S5 in a music class and for them there are five seats R1, R2, R3, R4 and R5 arranged in a row, where initially the seat Ri is allotted to the student Si, i = 1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats.

(There are two questions based on Paragraph "A", the question given below is one of them)

The probability that, on the examination day, the student S1 gets the previously allotted seat R1, and NONE of the remaining students gets the seat previously allotted to him/her is
A
$${3 \over {40}}$$
B
$${1 \over 8}$$
C
$${7 \over 40}$$
D
$${1 \over 5}$$
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