Probability – JEE Advanced Previous Year Questions with Solutions

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Let $\mathrm{U}_{1}$ and $\mathrm{U}_{2}$ be two urns such that $\mathrm{U}_{1}$ contains 3 white and 2 red balls, and $\mathrm{U}_{2}$ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from $\mathrm{U}_{1}$ and put into $\mathrm{U}_{2} .$ However, if tail appears then 2 balls are drawn at random from $\mathrm{U}_{1}$ and put into $\mathrm{U}_{2} .$ Now 1 ball is drawn at random from $\mathrm{U}_{2} .$

A box $\mathrm{B}_{1}$ contains 1 white ball, 3 red balls and 2 black balls. Another box $\mathrm{B}_{2}$ contains white balls, 3 red balls and 4 black balls. A third box $B_{3}$ contains 3 white balls, 4 red balls and 5 black balls.

Box 1 contains three cards bearing numbers, 1,2,3 ; box 2 contains five cards

bearing numbers 1,2,3,4,5; and box 3 contains seven cards bearing numbers 1,2,3,4,5,6,7. A card is drawn from each of the boxes. Let xi be the number on the card drawn from the ith box, i = 1,2,3.

Let $n_{1}$ and $n_{2}$ be the number of red and black balls respectively, in box I. Let $n_{3}$ and $n_{4}$ be the number of red and black balls, respectively, in box II.

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Football teams $\mathrm{T}_{1}$ and $\mathrm{T}_{2}$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $\mathrm{T}_{1}$ winning, drawing and losing a game against $\mathrm{T}_{2}$ are $\frac{1}{2}, \frac{1}{6}$ and $\frac{1}{3},$ respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $\mathrm{T}_{1}$ and $\mathrm{T}_{2},$ respectively, after two games

There are five students $S_{1}, S_{2}, S_{4}$ and $S_{5}$ in a music class and for them there are five sets $\mathrm{R}_{1}, \mathrm{R}_{2}, \mathrm{R}_{3}, \mathrm{R}_{4}$ and $\mathrm{R}_{5}$ arranged in a row, where initially the seat $\mathrm{R}_{\mathrm{i}}$ is allotted to the student $S_{i}, i=1,2,3,4,5 .$ 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)