Probability XII — JEE Advanced PYQs

Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red balls and 12 green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let ${\mathrm{E}}_1$ be the event that the ball chosen belonged to Box I …

A bookshelf contains 6 distinct books of Mathematics and 5 distinct books of Physics. From these 11 books, 6 books are chosen at random. Let X be the absolute value of the difference between the number of Mathematics books chosen and the number of Physics books chosen. If …

Three students $S_1,S_2$ and $S_3$ are given a problem to solve. Consider the following events :
$U:$ At least one of $S_1,S_2$ and $S_3$ can solve the problem,
$V:S_1$ can solve the problem, given that neither ${\mathrm{S}}_2$ nor …

A factory has a total of three manufacturing units, $M_1,M_2$, and $M_3$, which produce bulbs independent of each other. The units $M_1,M_2$, and $M_3$ produce bulbs in the proportions of 2: 2: 1, respectively. It is known that 20% of the bulbs produced in …

A bag contains $N$ balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For $i=1,2,3$, let $W_i,G_i$, and $B_i$ deno …

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct …

Let X be a random variable, and let $\mathrm{P}(\mathrm{X}=\mathrm{x})$ denote the probability that X takes the value x. Suppose that the points ( $\mathrm{x},\mathrm{P}(\mathrm{X}=\mathrm{x})$ ), $\mathrm{x}=0,1,2,3,4$, lie on a fixed straight line in the xy-plane, and …

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{3}$, then the probability that the experiment stops with head is

Let X be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in X while 02244 and 44422 are not in X . Suppose that each element of X has an equal chance of being chosen. Let p be the conditional probability that an element chosen at random is a mu …

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 …

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. I …

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 $b$ …

Consider three sets $E_1=\{1,2,3\},F_1=\{1,3,4\}$ and $G_1=\{2,3,4,5\}$. Two elements are chosen at random, without replacement, from the set $E_1$, and let $S_1$ denote the set of these chosen elements.
Let $E_2=E_1-S_1$ and $F_2=F_1\cup S_1$. Now two ele …

Let E, F and G be three events having probabilities
$P(E)=\frac{1}{8},P(F)=\frac{1}{6}$ and $P(G)=\frac{1}{4}$, and let $P(E\cap F\cap G)=\frac{1}{10}$.
For any event H, if $H^c$ denotes its complement, then which of the following statements is(are) TRUE?

Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting head in a single toss are $\frac{2}{3}$ and $\frac{1}{3}$, respectively. Suppose $\alpha$ is the number of heads that appear when $C_1$ is tossed twice, independently, and suppose $\beta$ is the number …

The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT …

Two fair dice, each with faces numbered 1, 2, 3, 4, 5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it tur …

There are three bags ${\mathrm{B}}_1,{\mathrm{B}}_2$ and ${\mathrm{B}}_3$. The bag ${\mathrm{B}}_1$ contains 5 red and 5 green balls, ${\mathrm{B}}_2$ contains 3 red and 5 green balls, and ${\mathrm{B}}_3$ contains 5 red and 3 green balls, Bags ${\mathrm{B}}_1,{\mathrm{B}}_2$ …

Let S be the sample space of all 3 × 3 matrices with entries from the set $\{0,1\}$. Let the events ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ be given by …

Let |X| denote the number of elements in set X. Let $\mathrm{S}=\{1,2,3,4,5,6\}$ be a sample space, where each element is equally likely to occur. If A and B are indepenent events associated with S, then the number of ordered pairs (A,B) such that …

Let $X$ and $Y$ be two events such that $P(X)=\frac{1}{3},P(X\mid Y)=\frac{1}{2}$ and $P(Y\mid X)=\frac{2}{5}$. Then

A computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces 20% and plant $T_2$ produces 80% of the total computers produced. 7% of computers produced in the factory turn out to be defective. It is known that
$P$ (computer turns out to be defect …

Football teams $T_1$ and $T_2$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_1$ winning, drawing and losing a game against $T_2$ are $\frac{1}{2},\frac{1}{6}$ and …

PARAGRAPH 1
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.

One of the two boxes, box I and box II, was selected at random and a ball was drawn ra …

PARAGRAPH 1
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.

A ball is drawn at random from box I and transferred to box II. If the probability of …

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is

Four persons independently solve a certain problem correctly with probabilities $\frac{1}{2},\frac{3}{4},\frac{1}{4},\frac{1}{8}$. Then the probability that the problem is solved correctly by at least one of them is

Of the three independent events $E_1,E_2$ and $E_3$, the probability that only $E_1$ occurs is $\alpha$, only $E_2$ occurs is $\beta$ and only $E_3$ occurs is $\gamma$. Let the probability $p$ that none of events $E_1,E_2$ or $E_3$ occurs …

A box $B_1$ contains 1 white ball, 3 red balls and 2 black balls. Another box $B_2$ contains 2 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.

If 1 ball is drawn from each of the boxes $B_1,B_2$ and …

A box $B_1$ contains 1 white ball, 3 red balls and 2 black balls. Another box $B_2$ contains 2 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.

If 2 balls are drawn (without replacement) from a randomly se …

Four fair dice $D_1,D_2,D_3$ and $D_4$, each having six faces numbered $1,2,3,4,5$ and 6, are rolled simultaneously. The probability that $D_4$ shows a number appearing on one of $D_1,D_2$ and $D_3$ is

Let $X$ and $Y$ be two events such that $P(X\mid Y)=\frac{1}{2},P(Y\mid X)=\frac{1}{3}$ and $P(X\cap Y)=\frac{1}{6}$. Which of the following is (are) correct?

A ship is fitted with three engines $E_1,E_2$ and $E_3$. The engines function independently of each other with respective probabilities $\frac{1}{2},\frac{1}{4}$ and $\frac{1}{4}$. For the ship to be operational at least two of its engines must function. Let $X$ denote th …

Let $U_1$ and $U_2$ be two urns such that $U_1$ contains 3 white and 2 red balls, and $U_2$ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from $U_1$ and put into $U_2$. However, if tail appears then 2 bal …

Let $U_1$ and $U_2$ be two urns such that $U_1$ contains 3 white and 2 red balls, and $U_2$ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from $U_1$ and put into $U_2$. However, if tail appears then 2 bal …

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\frac{11}{25}$ and the probability of none of them occurring is $\frac{2}{25}$. If $P(T)$ denotes the probability of occurrence of the event $T$, then

A signal which can be green or red with probability $\frac{4}{5}$ and $\frac{1}{5}$ respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is $\frac{3}{4}$. If the signal received at station …

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required.

The probability that $X=3$ equals

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required.

The probability that $X\ge 3$ equals

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required.

The conditional probability that $X\ge 6$ given $X>3$ equals

An experiment has 10 equally likely outcomes. Let $A$ and $B$ be two non-empty events of the experiment. If $A$ consists of 4 outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is

Let $H_1,H_2,\cdots ,H_n$ be mutually exclusive and exhaustive events with $P(H_i)>0$, $i=1,2,\dots ,n$. Let $E$ be any other event with $0<P(E)<1$.

STATEMENT-1 : $P(H_i\mid E)>P(E\mid H_i)\cdot P(H_i)$ for $i=1,2,\dots ,n$.

because

STATEME …

Let $E^{\mathrm{c}}$ denote the complement of an event $E$. Let $E,F,G$ be pairwise independent events with $P(G)>0$ and $P(E\cap F\cap G)=0$. Then $P\left(E^c\cap F^c\mid G\right)$ equals

There are n urns each containing $n+1$ balls such that the ith urn contains i white balls and $(n+1-i)$ red balls. Let $u_i$ be the event of selecting ith urn, $i=1,2,3\dots ,n$ and w denotes the event of getting a white ball.

If $P(u_i)\propto i$, where …

There are n urns each containing $n+1$ balls such that the ith urn contains i white balls and $(n+1-i)$ red balls. Let $u_i$ be the event of selecting ith urn, $i=1,2,3\dots ,n$ and w denotes the event of getting a white ball.

If $P(u_i)=c$, where c is a constant then …

There are n urns each containing $n+1$ balls such that the ith urn contains i white balls and $(n+1-i)$ red balls. Let $u_i$ be the event of selecting ith urn, $i=1,2,3\dots ,n$ and w denotes the event of getting a white ball.

If n is even and E denotes the event of choosing …