Suppose that

Box-I contains 8 red, 3 blue and 5 green balls,

Box-II contains 24 red, 9 blue and 15 green balls,

Box-III contains 1 blue, 12 green and 3 yellow balls,

Box-IV contains 10 green, 16 orange and 6 white balls.

A ball is chosen randomly from Box-I; call this ball $b$. If $b$ is red then a ball is chosen randomly from Box-II, if $b$ is blue then a ball is chosen randomly from Box-III, and if $b$ is green then a ball is chosen randomly from Box-IV. The conditional probability of the event 'one of the chosen balls is white' given that the event 'at least one of the chosen balls is green' has happened, is equal to

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:

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

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