Probability

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 ...

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 $E_1$ be the event that the ba...

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 $20 \%$ ...

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 $S_2$ nor $S_3$ can solve t...

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,...

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

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 ans...

Two distinct points are chosen randomly out of the points $A_1, A_2, \ldots, A_{49}$. Let $p$ be the probability that they are friends. Then the value of $7 p$ is :

Let $p_i$ be the probability that a randomly chosen point has $i$ many friends, $i=0,1,2,3,4$. Let $X$ be a random variable such that for $i=0,1,2,3,4$, the probability $P(X=i)=p_i$. Then the value of $7 E(X)$ 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 pro...

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 $X=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: \frac{x^2}{8}+\frac{y^2}{20}<1\right.$ and $\left.y^2<5 x\right\}$. Three distinct points $P, Q$ and $R$ are randomly chosen from $X$. Then the probability that $P, Q$ and $R$ form a tr...

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 ...

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 di...

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 coug...

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

Let E, F and G be three events having probabilities \(P(E) = {1 \over 8}\), \(P(F) = {1 \over 6}\) and \(P(G) = {1 \over 4}\), and let P (E \(\cap\) F \(\cap\) G) = \({1 \over {10}}\). For any event H, if Hc denotes the complement, then whi...

Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p1 be the probability that the maximum of chosen numbers is at least 81 and p2 be the probability that the minimum of chose...

Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p1 be the probability that the maximum of chosen numbers is at least 81 and p2 be the probability that the minimum of chose...

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 \(\cup\)...

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 ...

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 ...

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, indepe...

Let S be the sample space of all 3 \(\times\) 3 matrices with entries from the set {0, 1}. Let the events E1 and E2 be given byE1 = {A\(\in\)S : det A = 0} andE2 = {A\(\in\)S : sum of entries of A is 7}.If a matrix is chosen at random...

There are three bags B1, B2 and B3. The bag B1 contains 5 red and 5 green balls, B2 contains 3 red and 5 green balls, and B3 contains 5 red and 3 green balls. Bags B1, B2 and B3 have probabilities \({3 \over {10}}\), \({3 \over {10}}\) and ...

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 ...

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 ...

Three randomly chosen nonnegative integers x, y and z are found to satisfy the equation x + y + z = 10. Then the probability that z is even, is

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

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 $${1...

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 $${1...

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 defe...

Let \({n_1}\) and \({n_2}\) be the number of red and black balls, respectively, in box \({\rm I}\). Let \({n_3}\) and \({n_4}\) be the number of red and black balls, respectively, in box \({\rm I}{\rm I}.\)
A ball is drawn at random from b...

Let \({n_1}\) and \({n_2}\) be the number of red and black balls, respectively, in box \({\rm I}\). Let \({n_3}\) and \({n_4}\) be the number of red and black balls, respectively, in box \({\rm I}{\rm I}.\)
One of the two boxes, box $${\rm...

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

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}...

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}...

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...

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...

Four persons independently solve a certain problem correctly with probabilities \({1 \over 2},{3 \over 4},{1 \over 4},{1 \over 8}.\) 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 e...

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 ...

Let \(X\) and \(Y\) be two events such that \(P\left( {X|Y} \right) = {1 \over 2},\) \(P\left( {Y|X} \right) = {1 \over 3}\) and \(P\left( {X \cap Y} \right) = {1 \over 6}.\) 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 \({1 \over 2},{1 \over 4}\) and \({1 \over 4}\). For the ship to be operational at least two o...

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

Given that the drawn ball from \({U_2}\) is white, the probability that head appeared on the coin is

The probability of the drawn ball from \({U_2}\) being white is

A signal which can be green or red with probability \({4 \over 5}\) and \({1 \over 5}\) respectively, is received by station A and then transmitted to station \(B\). The probability of each station receving the signal correctly is $${3 \ove...

Let \(\omega\) be a complex cube root of unity with \(\omega \ne 1.\) A fair die is thrown three times. If \({r_1},\) \({r_2}\) and \({r_3}\) are the numbers obtained on the die, then the probability that $${\omega ^{{r_1}}} + {\omega ^{{...