Probability — JEE Advanced PYQs

Consider the $6\times 6$ square in the figure. Let ${\mathrm{A}}_1,{\mathrm{A}}_2,\dots ,{\mathrm{A}}_{49}$ be the points of intersections (dots in the picture) in some order. We say that ${\mathrm{A}}_i$ and ${\mathrm{A}}_j$ are friends if they are adjacent along a row …

Let $\mathrm{X}=\{(x,y)\in \mathbb{Z}\times \mathbb{Z}:\frac{x^2}{8}+\frac{y^2}{20}<1$ and $y^2<5x\}$. Three distinct points P, Q and R are randomly chosen from X. Then the probability that P, Q and R form a triangle whose area is a positive integer, is

In a study about a pandemic, data of 900 persons was collected. It was found that
190 persons had symptom of fever,
220 persons had symptom of cough,
220 persons had symptom of breathing problem,
330 persons had symptom of fever or cough or both,
350 persons had symptom of cough …

Three numbers are chosen at random, one after another with replacement, from the set $S=\{1,2,3,\dots ,100\}$. Let $p_1$ be the probability that the maximum of chosen numbers is at least 81 and $p_2$ be the probability that the minimum of chosen numbers is at most 40.

The …

Three numbers are chosen at random, one after another with replacement, from the set $S=\{1,2,3,\dots ,100\}$. Let $p_1$ be the probability that the maximum of chosen numbers is at least 81 and $p_2$ be the probability that the minimum of chosen numbers is at most 40.

The …

A number is chosen at random from the set $\{1,2,3,\dots ,2000\}$. Let p be the probability that the chosen number is a multiple of 3 or a multiple of 7 . Then the value of 500 p is ____.

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is

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 $x_i$ be the number on the card drawn from the $i^{\text{th }}$ …

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 $x_i$ be the number on the card drawn from the $i^{\text{th }}$ …

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is