Probability & Stats

Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained. The expectation of $X$ is ___________ (rounded off to two decimal places).

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement. What is the probability that the two balls drawn have different colours?

Suppose $X$ and $Y$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0,1]$. The probability that $X \geq Y$ is _______ .

In a class of 100 students, (i) there are 30 students who neither like romantic movies nor comedy movies, (ii) the number of students who like romantic movies is twice the number of students who like comedy movies, and (...

Out of 1000 individuals in a town, 100 unidentified individuals are covid positive. Due to lack of adequate covid-testing kits, the health authorities of the town devised a strategy to identify these covid-positive indiv...

A box contains the following three coins. I. A fair coin head on one face and tail on the other face. II. A coin with heads to both the faces. III. A coin with tails on both the faces. A coin is picked randomly from the...

Two continuous random variables X and Y are related as Y = 2X + 3. Let $$\sigma _x^2$$ and $$\sigma _y^2$$ denote the variances of X and Y, respectively. The variances are related as

The two sides of a fair coin are labelled as 0 and 1 . The coin is tossed two times independently. Let $M$ and $N$ denote the labels corresponding to the outcomes of those tosses. For a random variable $X$, defined as $X...

A digital communication system transmits a block of $N$ bits. The probability of error in decoding a bit is $\alpha$. The error event of each bit is independent of the error events of the other bits. The received block i...

The global financial crisis in 2008 is considered to be the most serious world-wide financial crisis, which started with the sub-prime lending crisis in USA in 2007. The sub-prime lending crisis led to the banking crisis...

$X$ is a random variable with uniform probability density function in the interval $[-2,10]$. For $Y=2 X-6$, the conditional probability $P(Y \leq 7 \mid X \geq 5)$ (rounded off to three decimal places) is $\_\_\_\_$ .

Five different books (P, Q, R, S, T) are to be arranged on a shelf. The books R and S are to be arranged first and second respectively from right side of the shelf. The number of different orders in which P, Q and T may...

A cab was involved in a hit and run accident at night. You are given the following data about the cabs in the city and the accident. (i) 85% of cabs in the city are green and the remaining cabs are blue. (ii) A witness i...

Let X 1 , X 2 , X 3 and X 4 be independent normal random variables with zero mean and unit variance. The probability that X 4 is the smallest among the four is _______.

If the number 715 ■ 423 is divisible by 3 (■ denotes the missing digit in the thousandths place), then the smallest whole number in the place of ■ is _______.

500 students are taking one or more courses out of Chemistry, Physics, and Mathematics. Registration records indicate course enrolment as follows: Chemistry (329), Physics (186), Mathematics (295), Chemistry and Physics...

The number of 3 digit numbers such that the digit 1 is never to the immediate right of 2 is

Three fair cubical dice are thrown simultaneously. The probability that all three dice have the same number of dots on the faces showing up is (up to third decimal place) _________.

$$500$$ students are taking one or more courses out of chemistry, physics and Mathematics. Registration records indicate course enrolment as follows: chemistry $$(329)$$, physics $$(186)$$, Mathematics $$(295)$$, chemist...

Passengers try repeatedly to get a seat reservation in any train running between two stations until they are successful. If there is $$40$$% chance of getting reservation in any attempt by a passenger, then the average n...

There are 3 Indians and 3 Chinese in a group of 6 people. How many subgroups of this group can we choose so that every subgroup has at least one Indian?

The second moment of a Poisson-distributed random variables is $$2.$$ The mean of the random variable is _______.

S, M, E and F are working in shifts in a team to finish a project. M works with twice the efficiency of others but for half as many days as E worked. S and M have 6 hour shifts in a day, whereas E and F have 12 hours shi...

Two random variables $$X$$ and $$Y$$ are distributed according to $$${f_{X,Y}}\left( {x,y} \right) = \left\{ {\matrix{ {\left( {x + y} \right),} & {0 \le x \le 1,} & {0 \le y \le 1} \cr {0,} & {otherwise} & \, \cr } } \r...

The bit error probability of a memoryless binary symmetric channel is $${10^{ - 5}}$$. If $${10^{ - 5}}$$ bits are sent over this channel, then the probability that not more than one bit will be in error is _____________...

The probability of getting a ''head'' in a single toss of a biased coin is $$0.3.$$ The coin is tossed repeatedly till a ''head'' is obtained. If the tosses are independent, then the probability of getting ''head'' for t...

The variance of the random variable $$X$$ with probability density function $$\,f\left( x \right) = {1 \over 2}\left| x \right|{e^{ - \left| x \right|}}\,\,$$ is ___________.

Let the random variable $$X$$ represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of $$X$$ is ________.

Suppose $$A$$ & $$B$$ are two independent events with probabilities $$P\left( A \right) \ne 0$$ and $$P\left( B \right) \ne 0.$$ Let $$\overrightarrow A $$ & $$\overrightarrow B $$ be their complements. Which of the foll...

Ram and Ramesh appeared in an interview for two vacancies in the same department. The probability of Ram's selection is $$1/6$$ and that of Ramesh is $$1/8$$. What is the probability that only one of them will be selecte...

Let $$X \in \left\{ {0,1} \right\}$$ and $$Y \in \left\{ {0,1} \right\}$$ be two independent binary random variables. If $$P\left( {X\,\, = 0} \right)\,\, = p$$ and $$P\left( {Y\,\, = 0} \right)\,\, = q,$$ then $$P\left(...

Let $$\,\,X \in \left\{ {0,1} \right\}\,\,$$ and $$\,\,Y \in \left\{ {0,1} \right\}\,\,$$ be two independent binary random variables. If $$\,\,P\left( {X\,\, = 0} \right) = p\,\,$$ and $$\,\,P\left( {Y\,\, = 0} \right) =...

Ram and Ramesh appeared in an interview for two vacancies in the same department. The probability of Ram’s selection is 1/6 and that of Ramesh is 1/8. What is the probability that only one of them will be selected?

A fair die with faces $$\left\{ {1,2,3,4,5,6} \right\}$$ is thrown repeatedly till $$'3'$$ is observed for the first time. Let $$X$$ denote the number of times the dice is thrown. The expected value of $$X$$ is _________...

You are given three coins: one has heads on both faces, the second has tails on both faces, and the third has a head on one face and a tail on the other. You choose a coin at random and toss it, and it comes up heads. Th...

If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier of future calls, the probability distribution function of the total number of calls in a...

If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier or future calls, the probability distribution function of the total number of calls in a...

In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random, has a sibling is _______.

Let $$X$$ be a random variable which is uniformly chosen from the set of positive odd numbers less than $$100.$$ The expectation, $$E\left[ X \right],$$ is __________.

Let $$X$$ be a zero mean unit variance Gaussian random variable. $$E\left[ {\left| X \right|} \right]$$ is equal to ______

An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is

Let $${X_1},\,{X_2},$$ and $${X_3}$$ be independent and identically distributed random variables with the uniform distribution on $$\left[ {0,\,1} \right]$$. The probability $$P\left\{ {{X_1} + {X_2} \le {X_3}} \right\}$...

Let $$X$$ be a real - valued random variable with $$E\left[ X \right]$$ and $$E\left[ {{X^2}} \right]$$ denoting the mean values of $$X$$ and $${{X^2}}$$, respectively. The relation which always holds true is

A fair coin is tossed repeatedly till both head and tail appear at least once. The average number of tosses required is ________.

Let $$\,{X_1},\,\,{X_2}\,\,$$ and $$\,{X_3}\,$$ be independent and identically distributed random variables with the uniform distribution on $$\left[ {0,1} \right].$$ The probability $$p$$ {$${X_1}$$ is the largest} is _...

Parcels from sender $$S$$ to receiver $$R$$ pass sequentially through two post - offices. Each post - office has a probability $${1 \over 5}$$ of losing an incoming parcel, independently of all other parcels. Given that...

Let $${X_1},{X_{2,}}$$ and $${X_{3,}}$$ be independent and identically distributed random variables with the uniform distribution on $$\left[ {0,1} \right].$$ The probability $$P\left\{ {{X_1} + {X_2} \le {X_3}} \right\}...

Let U and V be two independent zero mean Gaussian random variables of variances $${{1 \over 4}}$$ and $${{1 \over 9}}$$ respectively. The probability $$P(\,3V\, \ge \,\,2U)$$ is

What is the chance that a leap year, selected at random, will contain 53 Sundays?

Consider two identically distributed zero - mean random variables $$U$$ and $$V.$$ Let the cumulative distribution functions of $$U$$ and $$2V$$ be $$F(x)$$ and $$G(x)$$ respectively. Then for all values of $$x$$

Let $$U$$ and $$V$$ be two independent zero mean Gaussian random variables of variances $${1 \over 4}$$ and $${1 \over 9}$$ respectively. The probability $$\,P\left( {3V \ge 2U} \right)\,\,$$ is

Two independent random variable X and Y are uniformly distributed in the interval [ - 1, 1]. The probability that max [X, Y] is less than 1/2 is

A and B are friends. They decide to meet between 1 PM and 2 PM on a given day. There is a condition that whoever arrives first will not wait for the other for more than 15 minutes. The probability that they will meet on...

A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, is

Two independent random variables $$X$$ and $$Y$$ are uniformly distributed in the interval $$\left[ { - 1,1} \right].$$ The probability that max $$\left[ {X,Y} \right]$$ is less than $$1/2$$ is

There are eight bags of rice looking alike, seven of which have equal weight and one is slightly heavier. The weighing balance is of unlimited capacity. Using this balance, the minimum number of weighings required to ide...

A fair dice is tossed two times. The probability that the $$2$$ nd toss results in a value that is higher than the first toss is

A fair coin is tossed independently four times. The probability of the event ''The number of times heads show up is more than the number of times tails show up'' is

25 persons are in a room. 15 of them play hockey, 17 of them play football and 10 of them play both hockey and football. Then the number of persons playing neither hockey nor football is:

A fair coin is tossed $$10$$ times. What is the probability that only the first two tosses will yield heads?

Consider two independent random variables $$X$$ and $$Y$$ with identical distributions. The variables $$X$$ and $$Y$$ take values $$0, 1$$ and $$2$$ with probability $$1/2,$$ $$1/4$$ and $$1/4$$ respectively. What is the...

If $$E$$ denotes expectation, the variance of a random variable $$X$$ is given by

During transmission over a certain binary communication channel, bit errors occurs independently with probability p. The probability of at most one bit in error in a block of n bits is given by

If E denotes expectation, the variance of a random variable X is given by

An examination consists of two papers, paper $$1$$ and paper $$2.$$ The probability of failing in paper $$1$$ is $$0.3$$ and that in paper $$2$$ is $$0.2.$$ Given that a student has failed in paper $$2,$$ the probability...

A fair dice is rolled twice. The probability that an odd number will follow an even number is

During transmission over a communication channel, bit errors occur independently with probability 'p'. If a block of n bits is transmitted, the probability of at most one bit error is equal to

The PDF of a Gaussian random variable X is given by $${p_x}(x) = \,{1 \over {3\sqrt {2\pi } }}\,\exp \,[ - \,{(x - 4)^2}/18]$$. The probability of the event {X = 4} is

A probability density function is given by $$p(x) = \,K\,\,\exp \,\,( - \,{x^2}/2),\,\, - \,\infty \, < \,x < \,\infty $$. The value of K should be

In a digital communication system, transmissions of successive bits through a noisy channel are assumed to be independent events with error probability p. The probability of at most one error in the transmission of an 8...

The variance of a random variable X is $$\sigma _x^2\,.$$ Then the variance of - kx (where k is a positive constant ) is