Probability and Statistics · Engineering Mathematics · GATE ECE

Marks 1

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?

GATE ECE 2025

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 _______ .

GATE ECE 2024

The bar graph shown the frequency of the number of wickets taken in a match by a bowler in her career. For example, in 17 of her matches, the bowler has taken 5 wickets each. The median number of wickets taken by the bowler in a match is ____________ (rounded off to one decimal place).

GATE ECE 2022

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

GATE ECE 2021

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 $X=\min (M, N)$, the expected value $E[X]$ (rounded off to two decimal places) is $\_\_\_\_$ .

GATE ECE 2020

Let X₁ , X₂ , X₃ and X₄ be independent normal random variables with zero mean and unit variance. The probability that X₄ is the smallest among the four is _______.

GATE ECE 2018

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) _________.

GATE ECE 2017 Set 1

$$500$$ 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 $$(83)$$, chemistry and Mathematics $$(217)$$, and physic and Mathematics $$(63)$$, How many students are taking all $$3$$ subjects?

GATE ECE 2017 Set 2

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

GATE ECE 2016 Set 1

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 ___________.

GATE ECE 2015 Set 3

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 following statements is FALSE?

GATE ECE 2015 Set 1

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 ________.

GATE ECE 2015 Set 2

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?

GATE ECE 2015 Set 2

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

GATE ECE 2014 Set 1

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 _______.

GATE ECE 2014 Set 1

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

GATE ECE 2014 Set 3

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

GATE ECE 2014 Set 4

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 fixed time interval will be

GATE ECE 2014 Set 4

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

GATE ECE 2012

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

GATE ECE 2011

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

GATE ECE 2009

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

GATE ECE 2007

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

GATE ECE 2005

Marks 2

Let $X, N, Y$ and $Z$ be random variables. The variables $X$ and $N$ are independent of each other. $X$ is uniformly distributed between -1 and $1 ; N$ follows Normal distribution with zero mean and unity variance.

$Y$ and $Z$ are defined as, $Y=X+N$ and $Z=X^2+N$.

Which of the following pairs represents the values of correlation between $X$ and $Y$ and that between $X$ and $Z$ ?

GATE ECE 2026

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).

GATE ECE 2025

In a high school having equal number of boy students and girl students, 75% of the students study Science and the remaining 25% students study Commerce. Commerce students are two times more likely to be a boy than are Science students. The amount of information gained in knowing that a randomly selected girl student studies Commerce (rounded off to three decimal place) is ______ bits.

GATE ECE 2021

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 box and tossed. Out of the two remaining coins in the box, one coin is then picked randomly and tossed. If the first toss results in a head, the probability of getting a head in the second toss is

GATE ECE 2021

$X$ 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 $\_\_\_\_$ .

GATE ECE 2020

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 number of attempts that passengers need to make to get a seat reserved is ___________.

GATE ECE 2017 Set 2

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

The probability $$P\left( {X + Y \le 1} \right)$$ is ________.

GATE ECE 2016 Set 1

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 the first time in the fifth toss is ________.

GATE ECE 2016 Set 3

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 _________.

GATE ECE 2015 Set 3

The input $$X$$ to the Binary Symmetric Channel (BSC) shown in the figure is $$'1'$$ with probability $$0.8.$$ The cross-over probability is $$1/7$$. If the received bit $$Y=0,$$ the conditional probability that $$'1'$$ was transmitted is _______.