Probability & Counting

An urn contains one red ball and one blue ball. At each step, a ball is picked uniformly at random from the urn, and this ball together with another ball of the same color is put back in the urn. The probability that the...

Combinatorics deals with problems involving counting. For example, "How many distinct arrangements of N distinct objects in M spaces on a circle are possible?" is a typical problem in combinatorics. This kind of counting...

An unbiased six-faced dice whose faces are marked with numbers $1,2,3,4,5$, and 6 is rolled twice in succession and the number on the top face is recorded each time. The probability that the number appearing in the secon...

Let $X$ be a random variable which takes values in the set $\{1,2,3,4,5,6,7,8\}$. Further, $\operatorname{Pr}(X=1)=\operatorname{Pr}(X=2)=\operatorname{Pr}(X=5)=\operatorname{Pr}(X=7)=\frac{1}{6}$ and $\operatorname{Pr}(...

A day can only be cloudy or sunny. The probability of a day being cloudy is 0.5 , independent of the condition on other days. What is the probability that in any given four days, there will be three cloudy days and one s...

An unbiased six-faced dice whose faces are marked with numbers $1,2,3,4,5$, and 6 is rolled twice in succession and the number on the top face is recorded each time. The probability that the sum of the two recorded numbe...

The probability density function $f(x)$ of a random variable $X$ which takes real values is $$ f(x)=\frac{1}{3 \sqrt{2 \pi}} \exp \left(-\frac{x^2}{18}\right), x \in(-\infty,+\infty) $$ Which one of the following stateme...

Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all previous coin tosses. Let $E_1$ be the event that among the second, fourth, and sixth coin tosses, there are at least two heads. Let $E_2$...

A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, the probability $P($ head $)=0.5$ and for a fake coin, $P($ head $)=1$. You pick a coin at random and toss it twice, and get two hea...

The unit interval $(0,1)$ is divided at a point chosen uniformly distributed over $(0,1)$ in $R$ into two disjoint subintervals. The expected length of the subinterval that contains 0.4 is _________ . (rounded off to two...

Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. Flipping of each bit is independent of on...

Consider a probability distribution given by the density function $P(x)$. $$P(x)=\left\{\begin{array}{cc} C x^2, & \text { for } 1 \leq x \leq 4 \\ 0, & \text { for } x 4 \end{array}\right.$$ The probability that $x$ lie...

A fair six-faced dice, with the faces labelled ' 1 ', ' 2 ', ' 3 ', ' 4 ', ' 5 ', and ' 6 ', is rolled thrice. What is the probability of rolling ' 6 ' exactly once?

A quadratic polynomial $(x-\alpha)(x-\beta)$ over complex numbers is said to be square invariant if $(x-\alpha)(x-\beta)=\left(x-\alpha^2\right)\left(x-\beta^2\right)$. Suppose from the set of all square invariant quadra...

Consider the following sample of numbers: 9, 18, 11, 14, 15, 17, 10, 69, 11, 13 The median of the sample is

Let A and B be two events in a probability space with $P(A) = 0.3$, $P(B) = 0.5$, and $P(A \cap B) = 0.1$. Which of the following statements is/are TRUE?

When six unbiased dice are rolled simultaneously, the probability of getting all distinct numbers (i.e., 1, 2, 3, 4, 5, and 6) is

Let $ x $ and $ y $ be random variables, not necessarily independent, that take real values in the interval $[0,1]$. Let $ z = xy $ and let the mean values of $ x, y, z $ be $ \bar{x} , \bar{y} , \bar{z} $, respectively....

Consider a permutation sampled uniformly at random from the set of all permutations of {1, 2, 3, ..., n } for some n ≥ 4. Let X be the event that 1 occurs before 2 in the permutation, and Y the event that 3 occurs before...

A bag contains 10 red balls and 15 blue balls. Two balls are drawn randomly without replacement. Given that the first ball drawn is red, the probability (rounded off to 3 decimal places) that both balls drawn are red is...

Consider a random experiment where two fair coins are tossed. Let A be the event that denotes HEAD on both the throws, B be the event that denotes HEAD on the first throw, and C be the event that denotes HEAD on the seco...

A survey for a certain year found that 90% of pregnant women received medical care at least once before giving birth. Of these women, 60% received medical care from doctors, while 40% received medical care from other hea...

A box contains five balls of same size and shape. Three of them are green coloured balls and two of them are orange coloured balls. Balls are drawn from the box one at a time. If a green ball is drawn, it is not replaced...

For a given biased coin, the probability that the outcome of a toss is a head is 0.4. This coin is tossed 1,000 times. Let X denote the random variable whose value is the number of times that head appeared in these 1,000...

The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter 2. For a randomly picked component of this type, the probability that, its...

There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag. The probability that at least two chocolates are identical is ________.

Consider the two statements. S 1 : There exist random variables X and Y such that (E[X - E(X)) (Y - E(Y))]) 2 > Var[X] Var[Y] S 2 : For all random variables X and Y, Cov[X, Y] = E [|X - E[X]| |Y - E[Y]|] Which one of the...

In an examination, a student can choose the order in which two questions (QuesA and QuesB) must be attempted. - If the first question is answered wrong, the student gets zero marks. - If the first question is answered co...

A bag has r red balls and b black balls. All balls are identical except for their colours. In a trial, a ball is randomly drawn from the bag, its colour is noted and the ball is placed back into the bag along with anothe...

For n > 2, let a {0, 1} n be a non-zero vector. Suppose that x is chosen uniformly at random from {0, 1} n . Then, the probability that $$\sum\limits_{i = 1}^n {{a_i}{x_i}} $$ is an odd number is _______.

Let R be the set of all binary relations on the set {1,2,3}. Suppose a relation is chosen from R at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is _____.

Two numbers are chosen independently and uniformly at random from the set {1, 2, ...., 13}. The probability (rounded off to 3 decimal places) that their 4-bit (unsigned) binary representations have the same most signific...

Suppose Y is distributed uniformly in the open interval (1,6). The probability that the polynomial 3x 2 + 6xY + 3Y + 6 has only real roots is (rounded off to 1 decimal place) _____.

Two people, $$P$$ and $$Q,$$ decide to independently roll two identical dice, each with $$6$$ faces, numbered $$1$$ to $$6.$$ The person with the lower number wins. In case of a tie, they roll the dice repeatedly until t...

A six sided unbiased die with four green faces and two red faces is rolled seven times. Which of the following combinations is the most likely outcome of the experiment?

In a party, $$60\% $$ of the invited guests are male and $$400\% $$ are female. If $$80\% $$ of the invited guests attended the party and if all the invited female guests attended, what would be the ratio of males to fem...

The probability that a $k$-digit number does NOT contain the digits 0, 5, or 9 is :

$$P$$ and $$Q$$ are considering to apply for a job. The probability that $$P$$ applies for the job is $${1 \over 4},$$ the probability that $$P$$ applies for the job given that $$Q$$ applies for the job is $${1 \over 2},...

If a random variable $$X$$ has a Poisson distribution with mean $$5,$$ then the expectation $$E\left[ {{{\left( {X + 2} \right)}^2}} \right]$$ equals _________.

For any discrete random variable $$X,$$ with probability mass function $$P\left( {X = j} \right) = {p_j},$$ $${p_j}\,\, \ge 0,\,j \in \left\{ {0,..........,\,\,\,N} \right\},$$ and $$\,\,\sum\limits_{j = 0}^N {{p_j} = 1,...

Let $$X$$ be a Gaussian random variable with mean $$0$$ and variance $${\sigma ^2}$$ . Let $$Y=max(X,0)$$ where $$max(a, b)$$ is the maximum of $$a$$ and $$b$$. The median of $$Y$$ is ___________.

A probability density function on the interval $$\left[ {a,1} \right]$$ is given by $$1/{x^2}$$ and outside this interval the value of the function is zero. The value of $$a$$ is _________.

Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than $$100$$ hours given that it is of Type $$1$$ is $$0.7,$$ and given that it is of Type $$2$$ is...

Consider the following experiment. Step1: Flip a fair coin twice. Step2: If the outcomes are (TAILS, HEADS) then output $$Y$$ and stop. Step3: If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output $$N$...

Given Set $$\,\,\,A = \left\{ {2,3,4,5} \right\}\,\,\,$$ and Set $$\,\,\,B = \left\{ {11,12,13,14,15} \right\},\,\,\,$$ two numbers are randomly selected, one from each set. What is the probability that the sum of the tw...

The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p, and c respectively. Of these subjects, the student has 75% chance of passing in at least one, a 50% chance of passing in at least tw...

Let $$X$$ and $$Y$$ denote the sets containing $$2$$ and $$20$$ distinct objects respectively and $$𝐹$$ denote the set of all possible functions defined from $$X$$ to $$Y$$. Let $$f$$ be randomly chosen from $$F.$$ The...

Given set A = {2, 3, 4, 5} and Set B = {11, 12, 13, 14, 15}, two numbers are randomly selected, one from each set. What is probability that the sum of the two numbers equals 16?

The probabilities that a student passes in Mathematics, Physics and Chemistry are $$m, p$$ and $$c$$ respectively. Of these subjects, the student has $$75$$% chance of passing in at least one, a $$50$$% chance of passing...

Consider a software program that is artificially seeded with $$100$$ faults. While testing this program, $$159$$ faults are detected, out of which $$75$$ faults are from those artificially seeded faults. Assuming that bo...

Suppose $${X_i}$$ for $$i=1,2,3$$ are independent and identically distributed random variables whose probability mass functions are $$\,\,\Pr \left[ {{X_i} = 0} \right] = \Pr \left[ {{X_i} = 1} \right] = 1/2\,\,$$ for $$...

The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the system is functional, the officials inspect four of the computers picked at random (without replacem...

Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is X/1296. The value of X is__________

Let S be a sample space and two mutually exclusive events A and B be such that $$A\, \cup \,B = \,S$$. If P(.) denotes the probability of the event, the maximum value of P(A) P(B) is ________________.

The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is _______________

Each of the nine words in the sentence "The Quick brown fox jumps over the lazy dog" is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the bo...

Suppose you break a stick of unit length at a point chosen uniformaly at random. Then the expected length of the shorter stick is __________________.

Suppose p is the number of cars per minute passing through a certain road junction between 5PM and 6PM and p has a poisson distribution with mean 3. What is the probability of observing fewer than 3 cars during any given...

Consider an undirected random$$^ \circ $$ graph of eight vertices. The probability that there is an edge between a pair of vertices is 1/2. What is the expected number of unordered cycles of length three?

Suppose a fair six-sided die is rolled once. If the value on the die is 1, 2 or 3 the die is rolled a second time. What is the probability that the sum total of values that turn up is at least 6?

Consider a random variable X that takes values + 1 and-1 with probability 0.5 each. The values of the cumulative distribution function F(x) at x = - 1 and + 1 are

A deck of 5 cards (each carrying a distinct number from 1 to 5) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is probability that the two cards are selected with the number on the f...

If the difference between the expectation of the square of a random variable $$\left( {E\left[ {{X^2}} \right]} \right)$$ and the square of the expectation of the random variable $${\left( {E\left[ X \right]} \right)^2}$...

Consider a finite sequence of random values $$X = \left\{ {{x_1},{x_2},{x_3}, - - - - - {x_n}} \right\}..$$ Let $${\mu _x}$$ be the mean and $${\sigma _x}$$ be the standard deviation of $$X.$$ Let another finite sequence...

Consider a company that assembles computers. The probability of a faulty assembly of any computer is P. The company therefore subjects each computer to a testing process. This gives the correct result for any computer wi...

What is the probability that divisor of $${10^{99}}$$ is a multiple of $${10^{96}}$$ ?

An unbalanced dice (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of the probability that the face valueis even. The probability of getting any even bnumbered face is th...

What is the probability that in a randomly choosen group of r people at least three people have the same birthday?

Let X be a random variable following normal distribution with mean + 1 and variance 4. Let Y be another normal variable with mean - 1 and variance unknown. If $$P\,(X\, \le \, - 1) = \,P(Y\,\, \ge \,2)$$, the standard de...

A sample space has two events A and B such that probabilities $$P\,(A\, \cap \,B)\, = \,1/2,\,\,P(\overline A )\, = \,1/3,\,\,P(\overline B )\, = \,1/3$$. What is P $$P\,(A\, \cup \,B)\,$$?

Aishwarya studies either computer science or mathematics everyday. If she studies computer science on a day, then the probability that the studies mathematics the next day is 0.6. If she studies mathematics on a day, the...

Suppose there are two coins. The first coin gives heads with probability 5/8 when tossed, while the second coin gives heads with probability 1/4. On e of the two coins is picked up at random with equal probability and to...

Suppose we uniformly and randomly select a permutation from the 20! permutations of 1, 2, 3,..., 20. What is the promutations that 2 appears at an earlier position than any other even number in the selected permutation?

When a coin is tossed, the probability of getting a Head is p, 0 < p < 1. Let N be the random variable denoting the number of tosses till the first Head appears, including the toss where the Head appears. Assuming that s...

In a certain town, the probability that it will rain in the afternoon is known to be 0.6. Moreover, meteorological data indicates that if the temperature at noon is less than or equal to $${25^ \circ }$$ C, the probabili...

For each elements in a set of size $$2n$$, an unbiased coin in tossed. The $$2n$$ coin tosses are independent. An element is chhoosen if the corresponding coin toss were head.The probability that exactly $$n$$ elements a...

A random bit string of length n is constructed by tossing a fair coin n times and setting a bit to 0 or 1 depending on outcomes head and tail, respectively. The probability that two such randomly generated strings are no...

Let $$f(x)$$ be the continuous probability density function of a random variable X. The probability that $$a\, < \,X\, \le \,b$$, is:

An unbiased coin is tossed repeatedly until the outcome of two successive tosses is the same. Assuming that the tails are independent, the expected number of tosses are

A bag contains 10 blue marbles, 20 green marbles and 30 red marbles. A marble is drawn from the bag, its colour recorded and it is put back in the bag. This process is repeated 3 times. The probability that no two of the...

Box P has 2 red balls and 3 blue balls and box Q has 3 red balls and 1 blue ball. A ball is selected as follows: (i) select a box (ii) choose a ball from the selected box such that each ball in the box is equally likely...

Two n bit binary stings, S1 and, are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings different) is equal to d i...

If a fair coin is tossed four times, what is the probability that two heads and two tails will result?

A point is randomly selected with uniform probability in the X-Y plane within the rectangle with corners at (0, 0), (1, 0), (1, 2) and (0, 2). If p is the length of the position vector of the point, the expected value of...

In a population of N families, 50% of the families have three children, 30% of the families have two children and the remaining families have one child. What is the probability that a randomly picked child belongs to a f...

An examination paper has 150 multiple-choice questions of one mark each, with each question having four choices. Each incorrect answer fetches-0.25 mark. Suppose 1000 students choose all their answers randomly with unifo...

Let P(E) denote the probability of the event E. Given P(A) = 1, P(B) = $${\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$$, the values of $$P\,(A\,\left| {B) \,} \right.$...

Consider the following algorithm for searching for a given number x in an unsorted array A[1..n] having n distinct values: 1. Choose an i uniformly at random fro 1..n; 2. If A[i]=x then stop else Goto 1; Assuming that x...

Four fair coins are tossed simultaneously. The probability that at least one head and one tail turn up is

Seven (distinct) car accidents occurred in a week. What is the probability that they all occurred on the same day ?

$${{E_1}}$$ and $${{E_2}}$$ are events in a probability space satisfying the following constraints: $$ \bullet $$ $$\Pr \,\,({E_1}) = \Pr \,({E_2})$$ $$ \bullet $$ $$\Pr \,\,({E_1}\, \cup {E_2}) = 1$$ $$ \bullet $$ $${E_...

Suppose that the expectation of a random variable X is 5. Which of the following statements is true?

Consider two events $${{E_1}}$$ and $${{E_2}}$$ such that probability of $${{E_1}}$$, Pr [$${{E_1}}$$] = 1/2, probability of $${{E_2}}$$, Pr[$${{E_2}}$$ = 1/3, and probability of $${{E_1}}$$ and $${{E_2}}$$, $$\left[ {{E...

Suppose that the expectation of a random variable X is 5. Which of the following statements is true?

Let X and Y be two exponentially distributed and independent random variables with mean $$\alpha $$ and $$\beta $$, respectively. If Z = min (X, Y), then the mean of Z is given by

A die is rolled three times. The probability that exactly one odd number turns up among the three outcomes is

The probability that it will rain today is 0.5. The probability that it will rain tomorrow is 0.6. The probability that it will rain either today or tomorrow is 0.7. That is the probability that it will rain today and to...

The probability that the top and bottom cards of a randomly shuffled deck are both access is

Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is

The probability that a number selected at random between $$100$$ and $$999$$ (both inclusive ) will not contain the digit $$7$$ is

A bag contains 10 white balls and 15 black balls. Two balls drawn in succession. The probability that one of them is black the other is white is

Let A and B be any two arbitrary events, then, which one of the following is true?