Recent questions and answers in Probability / GATE Overflow for GATE CSE

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IIT Madras MS DSAI Written Test 2024
Two fair 6-face diced are tossed independently. Let X be the random variable of the sum of two numbers on dices and let Y be the absolute difference of two numbers on dices. What is the value of P( X $\geq$ 2Y)? a) 22/36 b) 24/36 c) 26/36 c) 28/36 d) 32/36

Two fair 6-face diced are tossed independently. Let X be the random variable of the sum of two numbers on dices and let Y be the absolute difference of two numbers on dic...

harshrajhrj

6 views

harshrajhrj asked 2 hours ago

3 votes

2 answers

2

GATE CSE 2024 | Set 1 | Question: 4
Consider a permutation sampled uniformly at random from the set of all permutations of $\{1,2,3, \cdots, n\}$ for some $n \geq 4$. Let $X$ be the event that $1$ occurs before $2$ in the permutation, and $Y$ the event that $3$ occurs before ... The events $X$ and $Y$ are independent Either event $X$ or $Y$ must occur Event $X$ is more likely than event $Y$

Consider a permutation sampled uniformly at random from the set of all permutations of $\{1,2,3, \cdots, n\}$ for some $n \geq 4$. Let $X$ be the event that $1$ occurs be...

Bhaskar_Saini

2.5k views

Bhaskar_Saini answered 1 day ago

6 votes

2 answers

3

GATE CSE 2024 | Set 1 | Question: 17
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? The two events $A$ and $B$ are independent $P(A \cup B)=0.7$ ... $B$ $P\left(A^c \cap B^c\right)=0.4$, where $A^c$ and $B^c$ are the complements of the events $A$ and $B$, respectively

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?The two events $A$ an...

Bhaskar_Saini

2.1k views

Bhaskar_Saini answered 1 day ago

1 votes

2 answers

4

GATE CSE 2024 | Set 2 | Question: 8
When six unbiased dice are rolled simultaneously, the probability of getting all distinct numbers $(i.e., 1, 2, 3, 4, 5, \text{and } 6)$ is $\frac{1}{324}$ $\frac{5}{324}$ $\frac{7}{324}$ $\frac{11}{324}$

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

Bhaskar_Saini

2.2k views

Bhaskar_Saini answered 1 day ago

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5

David Stirzaker, Elementary Probability, Chapter 1, Example 1.9 Urn
An urn contains $n$ heliotrope and $n$ tangerine balls. A fair die with $n$ sides is rolled. If the $r^{th}$ face is shown, $r$ balls are removed from the urn and placed in a bag. What is the probability that a ball removed at random from the bag is tangerine?

An urn contains $n$ heliotrope and $n$ tangerine balls. A fair die with $n$ sides is rolled. If the $r^{th}$ face is shown, $r$ balls are removed from the urn and placed ...

Priyam Garg

56 views

Priyam Garg asked 3 days ago

79 votes

12 answers

6

GATE CSE 2009 | Question: 21
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 value is even. The probability of getting any even numbered face is the same. If the ... following options is closest to the probability that the face value exceeds $3$? $0.453$ $0.468$ $0.485$ $0.492$

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 value is ev...

harshitraj12

16.7k views

harshitraj12 answered Apr 20

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2 answers

7

GATE DS&AI 2024 | Question: 48
Consider two events $T$ and $S$. Let $\bar{T}$ denote the complement of the event $T$. The probability associated with different events are given as follows: \[ P(\bar{T})=0.6, \quad P(S \mid T)=0.3, \quad P(S \mid \bar{T})=0.6 \] Then, $P(T \mid S)$ is $\_\_\_\_\_\_\_\_$ (rounded off to two decimal places).

Consider two events $T$ and $S$. Let $\bar{T}$ denote the complement of the event $T$. The probability associated with different events are given as follows:\[P(\bar{T})=...

Sahil5635

736 views

Sahil5635 answered Apr 13

46 votes

11 answers

8

GATE CSE 2016 Set 2 | Question: 05
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 $0.4$. The probability that an LED bulb chosen uniformly at random lasts more than $100$ hours 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 ...

Aman Kumar 21

9.7k views

Aman Kumar 21 answered Apr 12

0 votes

1 answer

9

Memory Based GATE DA 2024 | Question: 41
Consider two random variables, $x$ and $y$, defined as follows: \[ x = \begin{cases} 1 & \text{if HH } \\ 0 & \text{otherwise} \end{cases} \] \[ y = \begin{cases} 1 & \text{if at least one head} \\ 0 & \text{otherwise} \end{cases} \] What is the covariance between $x$ and $y?$

Consider two random variables, $x$ and $y$, defined as follows:\[x =\begin{cases}1 & \text{if HH } \\0 & \text{otherwise}\end{cases}\]\[y =\begin{cases}1 & \text{if at le...

nanilu

157 views

nanilu answered Apr 5

7 votes

3 answers

10

Mathematics GATE 2011 probability
A fair die is tossed two times. the probability that 2nd toss results in value greater than first toss is ?

A fair die is tossed two times. the probability that 2nd toss results in value greater than first toss is ?

Creatorpk

2.4k views

Creatorpk answered Mar 17

16 votes

3 answers

11

TIFR CSE 2014 | Part A | Question: 17
A fair dice (with faces numbered $1, . . . , 6$) is independently rolled repeatedly. Let $X$ denote the number of rolls till an even number is seen and let $Y$ denote the number of rolls till $3$ is seen. Evaluate $E(Y |X = 2)$. $6\frac{5}{6}$ $6$ $5\frac{1}{2}$ $6\frac{1}{3}$ $5\frac{2}{3}$

A fair dice (with faces numbered $1, . . . , 6$) is independently rolled repeatedly. Let $X$ denote the number of rolls till an even number is seen and let $Y$ denote the...

Priyam Garg

4.0k views

Priyam Garg answered Feb 25

19 votes

7 answers

12

TIFR CSE 2013 | Part A | Question: 6
You are lost in the National park of Kabrastan. The park population consists of tourists and Kabrastanis. Tourists comprise two-thirds of the population the park and give a correct answer to requests for directions with probability $\dfrac{3}{4}$. The air of Kabrastan has an ... $\left(\dfrac{1}{2}\right)$ $\left(\dfrac{2}{3}\right)$ $\left(\dfrac{3}{4}\right)$

You are lost in the National park of Kabrastan. The park population consists of tourists and Kabrastanis. Tourists comprise two-thirds of the population the park and give...

Priyam Garg

3.3k views

Priyam Garg answered Feb 25

9 votes

5 answers

13

TIFR CSE 2013 | Part A | Question: 17
A stick of unit length is broken into two at a point chosen at random. Then, the larger part of the stick is further divided into two parts in the ratio $4:3$. What is the probability that the three sticks that are left CANNOT form a triangle? $1/4$ $1/3$ $5/6$ $1/2$ $\log_{e}(2)/2$

A stick of unit length is broken into two at a point chosen at random. Then, the larger part of the stick is further divided into two parts in the ratio $4:3$. What is th...

Priyam Garg

1.9k views

Priyam Garg answered Feb 24

25 votes

7 answers

14

TIFR CSE 2011 | Part A | Question: 19
Three dice are rolled independently. What is the probability that the highest and the lowest value differ by $4$? $\left(\dfrac{1}{3}\right)$ $\left(\dfrac{1}{6}\right)$ $\left(\dfrac{1}{9}\right)$ $\left(\dfrac{5}{18}\right)$ $\left(\dfrac{2}{9}\right)$

Three dice are rolled independently. What is the probability that the highest and the lowest value differ by $4$? $\left(\dfrac{1}{3}\right)$ $\left(\dfrac{1}{6}\righ...

Priyam Garg

3.0k views

Priyam Garg answered Feb 23

23 votes

5 answers

15

TIFR CSE 2012 | Part A | Question: 20
There are $1000$ balls in a bag, of which $900$ are black and $100$ are white. I randomly draw $100$ balls from the bag. What is the probability that the $101$st ball will be black? $9/10$ More than $9/10$ but less than $1$. Less than $9/10$ but more than $0$. $0$ $1$

There are $1000$ balls in a bag, of which $900$ are black and $100$ are white. I randomly draw $100$ balls from the bag. What is the probability that the $101$st ball wil...

Priyam Garg

2.9k views

Priyam Garg answered Feb 19

2 votes

3 answers

16

GATE CSE 2024 | Set 2 | Question: 34
Let $x$ and $y$ be random variables, not necessarily independent, that take real values in the interval $[0,1]$. Let $z=x y$ and let the mean values of $x, y, z$ be $\bar{x}, \bar{y}, \bar{z}$ ... $\bar{z} \leq \bar{x} \bar{y}$ $\bar{z} \geq \bar{x} \bar{y}$ $\bar{z} \leq \bar{x}$

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

Argharupa Adhikary

2.8k views

Argharupa Adhikary answered Feb 18

2 votes

2 answers

17

GATE CSE 2024 | Set 1 | Question: 53
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 ___________.

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 ...

shishir__roy

2.3k views

shishir__roy answered Feb 16

1 votes

1 answer

18

GATE DS&AI 2024 | Question: 47
Let $X$ be a random variable exponentially distributed with parameter $\lambda>0$. The probability density function of $X$ is given by: \[ f_{X}(x)=\left\{\begin{array}{ll} \lambda e^{-\lambda x}, \quad x \geq 0 \\ 0, & \text { otherwise } \end ... variance of $X$, respectively, the value of $\lambda$ is $\_\_\_\_\_\_\_\_$ (rounded off to one decimal place).

Let $X$ be a random variable exponentially distributed with parameter $\lambda>0$. The probability density function of $X$ is given by:\[f_{X}(x)=\left\{\begin{array}{ll}...

NarutoUzumaki

706 views

NarutoUzumaki answered Feb 16

1 votes

1 answer

19

GATE DS&AI 2024 | Question: 1
Consider the following statements: The mean and variance of a Poisson random variable are equal. For a standard normal random variable, the mean is zero and the variance is one. Which ONE of the following options is correct? Both $\text{(i)}$ and $\text{(ii)}$ are true ... $\text{(i)}$ is false Both $\text{(i)}$ and $\text{(ii)}$ are false

Consider the following statements:The mean and variance of a Poisson random variable are equal.For a standard normal random variable, the mean is zero and the...

NarutoUzumaki

1.3k views

NarutoUzumaki answered Feb 16

1 votes

1 answer

20

GATE DS&AI 2024 | Question: 26
A fair six-sided die (with faces numbered $1,2,3,4,5,6$ ) is repeatedly thrown independently. What is the expected number of times the die is thrown until two consecutive throws of even numbers are seen? $2$ $4$ $6$ $8$

A fair six-sided die (with faces numbered $1,2,3,4,5,6$ ) is repeatedly thrown independently.What is the expected number of times the die is thrown until t...

ankitgupta.1729

1.2k views

ankitgupta.1729 answered Feb 16

2 votes

2 answers

21

JEST 2019
Three dice are rolled independently. Probability of obtaining the difference from largest and smallest number as exactly 4 :

Three dice are rolled independently. Probability of obtaining the difference from largest and smallest number as exactly 4 :

Priyam Garg

522 views

Priyam Garg answered Feb 11

2 votes

2 answers

22

GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 21
Suppose we have events $A, B$ in a sample space. And we know that $\mathrm{P}(\mathrm{A})=0.3, \mathrm{P}\left(\mathrm{B} \mid \mathrm{A}^c\right)=0.25, \mathrm{P}(\mathrm{B} \mid \mathrm{A})=0.45$. What is $\mathrm{P}\left(\mathrm{A}^c \mid \mathrm{B}\right) ?$ 0.75 0.55 0.2 0.56

Suppose we have events $A, B$ in a sample space. And we know that $\mathrm{P}(\mathrm{A})=0.3, \mathrm{P}\left(\mathrm{B} \mid \mathrm{A}^c\right)=0.25, \mathrm{P}(\mathr...

chidambareswar23

563 views

chidambareswar23 answered Feb 7

0 votes

1 answer

23

Memory Based GATE DA 2024 | Question: 42
Consider two random variables, (x) and (y), each following a uniform distribution. Specifically, (x) is uniformly distributed over the interval ([1, 3]), and (y) is uniformly distributed over the interval ([2, 4]). What will be $P(x \geq y)$

Consider two random variables, (x) and (y), each following a uniform distribution. Specifically, (x) is uniformly distributed over the interval ([1, 3]), and (y) is uni...

RahulVerma3

319 views

RahulVerma3 answered Feb 6

1 votes

2 answers

24

Memory Based GATE DA 2024 | Question: 13
A fair six-sided die is thrown repeatedly. Find the expected number of throws until two consecutive throws show even numbers.

A fair six-sided die is thrown repeatedly. Find the expected number of throws until two consecutive throws show even numbers.

Mumuksh29

700 views

Mumuksh29 answered Feb 5

0 votes

1 answer

25

Memory Based GATE DA 2024 | Question: 14
Consider two events $\mathrm{T}$ and $\mathrm{S}$. Let $\overline{\mathrm{T}}$ denote the complement of event $\mathrm{T}$. The probabilities associated with different events are given as follows: $\mathrm{P}({\mathrm{T}})=0.4$ ... $\mathrm{P}(\mathrm{T}|\mathrm{S})$.

Consider two events $\mathrm{T}$ and $\mathrm{S}$. Let $\overline{\mathrm{T}}$ denote the complement of event $\mathrm{T}$. The probabilities associated with different ev...

Mumuksh29

261 views

Mumuksh29 answered Feb 5

2 votes

1 answer

26

Memory Based GATE DA 2024 | Question: 1
Consider the probability density function $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$ and $\lambda > 0$. Determine the value of $\lambda$ such that $5E(x) = V(x)$.

Consider the probability density function $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$ and $\lambda 0$. Determine the value of $\lambda$ such that \(5E(x) = V(...

Mrityudoot

353 views

Mrityudoot answered Feb 5

2 votes

1 answer

27

GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 48
Consider the quadratic equation $x^2+\dfrac{x}{2}+c=0$, where $c$ is chosen uniformly randomly from the interval $[0,1]$. What is the probability that the given quadratic equation has a real solution? The solutions of $a x^2+b x+c=0$ are given by $x=\dfrac{-b \pm \sqrt{b^2-4 a c}}{2a}$. $1 / 2$ $1 / 4$ $1 / 8$ $1 / 16$

Consider the quadratic equation $x^2+\dfrac{x}{2}+c=0$, where $c$ is chosen uniformly randomly from the interval $[0,1]$. What is the probability that the given quadratic...

GO Classes

382 views

GO Classes answered Feb 5

0 votes

0 answers

28

Memory Based GATE DA 2024 | Question: 11
Consider a dataset with a raw score $x = 75$, a mean $\mu = 70$, and a standard deviation $\sigma = 5$. Calculate the Z-score using the formula $z = \frac{x - \mu}{\sigma}$.

Consider a dataset with a raw score $x = 75$, a mean $\mu = 70$, and a standard deviation $\sigma = 5$. Calculate the Z-score using the formula \(z = \frac{x - \mu}...

GO Classes

205 views

GO Classes asked Feb 4

1 votes

0 answers

29

Memory Based GATE DA 2024 | Question: 15
Consider the joint probability density function given by: $ f(x, y)= \begin{cases} 2xy & \text{if } 0 < x < 2 \text{ and } 0 < y < x \\ 0 & \text{otherwise} \end{cases} $ \noindent Determine the conditional expectation $E(Y | X = 1.5)$.

Consider the joint probability density function given by:$$f(x, y)=\begin{cases} 2xy & \text{if } 0 < x < 2 \text{ and } 0 < y < x \\ 0 & \text{otherwise}\e...

GO Classes

218 views

GO Classes asked Feb 4

0 votes

0 answers

30

Memory Based GATE DA 2024 | Question: 35
Conditional probability \[ \begin{aligned} & \mathrm{P}(\mathrm{U}, \mathrm{V}, \mathrm{W}, \mathrm{X}, \mathrm{Y}) & = \mathrm{P}(\mathrm{U}) \cdot \mathrm{P}(\mathrm{V}) \cdot \mathrm{P}(\mathrm{W} / \mathrm{U}, \mathrm{V}) \cdot \mathrm{P}(\mathrm{X} / \mathrm{W}) \cdot \mathrm{P}(\mathrm{Y} / \mathrm{W}) \end{aligned} \]

Conditional probability\[\begin{aligned} & \mathrm{P}(\mathrm{U}, \mathrm{V}, \mathrm{W}, \mathrm{X}, \mathrm{Y}) & = \mathrm{P}(\mathrm{U}) \cdot \mathrm{P...

GO Classes

166 views

GO Classes asked Feb 4

0 votes

0 answers

31

Memory Based GATE DA 2024 | Question: 44
Consider the statements below related to probability distributions: \textbf{(S1):} For a Poisson distribution, the mean and variance are equal. \textbf{(S2):} For a standard normal distribution, the mean is 0, and the variance is 1. Which of the following ... true. S1 is true, but S2 is false. S1 is false, but S2 is true. Both S1 and S2 are false.

Consider the statements below related to probability distributions:\textbf{(S1):} For a Poisson distribution, the mean and variance are equal.\textbf{(S2):} For a standar...

GO Classes

187 views

GO Classes asked Feb 4

2 votes

3 answers

32

Probability
In a certain group of computer personnel, 65% have insufficient knowledge of hardware, 45% have inadequate idea of software and 70% are in either one (or) both of the two categories. What is the percentage of people who knpw software among those who have a sufficient knowledge of hardware? 0.35/0.3 0.3/0.35 0.3 0.35

In a certain group of computer personnel, 65% have insufficient knowledge of hardware, 45% have inadequate idea of software and 70% are in either one (or) both of the two...

Sbrjt

2.5k views

Sbrjt answered Feb 4

6 votes

1 answer

33

GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 12
Let $x$ be a random variable possessing the probability density function $ f(x)= \begin{cases}c x & , x \in[0,10] \\ 0 & , \text { otherwise }\end{cases} $ where $c \in \mathbb{R}$. The probability that $x \in[1,2]$ is ______. $\dfrac{1}{100}$ $\dfrac{3}{100}$ $\dfrac{5}{100}$ $\dfrac{7}{100}$

Let $x$ be a random variable possessing the probability density function$$f(x)= \begin{cases}c x & , x \in[0,10] \\ 0 & , \text { otherwise }\end{cases}$$where $c \in \ma...

SankarVinayak

509 views

SankarVinayak answered Jan 29

4 votes

1 answer

34

GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 44
Football teams $T_1$ and $T_2$ play two games against each other in the Premier League. It is assumed that the outcomes of the two games are independent of each other. The probabilities of $T_1$ winning, drawing and losing against $T_2$ ... What will be the value of $P(X=Y)?$ $1 / 3$ $13 / 36$ $1 / 36$ $1 / 18$

Football teams $T_1$ and $T_2$ play two games against each other in the Premier League. It is assumed that the outcomes of the two games are independent of each other. Th...

GO Classes

486 views

GO Classes answered Jan 28

8 votes

2 answers

35

GO Classes Test Series 2024 | Mock GATE | Test 12 | Question: 6
A college has $10$ (non-overlapping) time slots for its courses, and assigns courses to time slots randomly and independently. A student randomly chooses $3$ of the courses to enroll in. What is the probability that there is a conflict in the student's schedule? (answer upto $2$ decimals)

A college has $10$ (non-overlapping) time slots for its courses, and assigns courses to time slots randomly and independently. A student randomly chooses $3$ of the cours...

krishnajsw

850 views

krishnajsw answered Jan 21

6 votes

1 answer

36

GO Classes Test Series 2024 | Mock GATE | Test 12 | Question: 36
Let $A, B, C$ be events such that $P(A)=P(B)=P(C)=0.5, P(A \cap B)=0.3, P(A \cap C)=0$. Which of the following is/are true? $P(A \cup B)=0.75$ $P(A \cup C)=1$ $P(B \cap C)=0.23$ $P(B \cup C)=0.9$

Let $A, B, C$ be events such that $P(A)=P(B)=P(C)=0.5, P(A \cap B)=0.3, P(A \cap C)=0$.Which of the following is/are true?$P(A \cup B)=0.75$$P(A \cup C)=1$$P(B \cap C)=0....

GO Classes

893 views

GO Classes answered Jan 21

3 votes

1 answer

37

GO Classes Test Series 2024 | Mock GATE | Test 11 | Question: 28
Suppose that $X$ and $Y$ are independent random variables such that each is equal to $0$ with probability $.5$ and $1$ with probability $.5.$ Find $P(X+Y \leq 1)?$ (Answer up to $2$ decimals)

Suppose that $X$ and $Y$ are independent random variables such that each is equal to $0$ with probability $.5$ and $1$ with probability $.5.$Find $P(X+Y \leq 1)?$ (Answe...

GO Classes

537 views

GO Classes asked Jan 13

4 votes

1 answer

38

GO Classes Test Series 2024 | Mock GATE | Test 11 | Question: 29
You have three coins in your pocket, two fair ones but the third biased with the probability of heads $p$ and tails $1-p$. One coin selected at random drops to the floor, landing heads up. How likely is it that it is one of the fair coins? $1 / \mathrm{p}$ $1 /(1+p)$ $p /(1+p)$ $(1+p) / p$

You have three coins in your pocket, two fair ones but the third biased with the probability of heads $p$ and tails $1-p$. One coin selected at random drops to the floor,...

GO Classes

481 views

GO Classes asked Jan 13

0 votes

0 answers

39

Madeeasy 2024 full length test 3
Can anyone please help me solve this question or explain the formula used in the solution part of the same?

Can anyone please help me solve this question or explain the formula used in the solution part of the same?

VinayBhojwani

171 views

VinayBhojwani asked Jan 12

0 votes

1 answer

40

GATE 2016 | MATHS | Q-33
Suppose $ X $ and $ Y $ are two random variables such that $ aX + bY $ is a normal random variable for all $ a, b $ in $ \mathbf{R} $. Consider the following statements P, Q, R, and S: (P) : $ X $ is a standard normal random variable. (Q) ... 0. Which of the above statements ALWAYS hold TRUE? (A) both P and Q (B) both Q and R (C) both Q and S (D) both P and S

Suppose $ X $ and $ Y $ are two random variables such that $ aX + bY $ is a normal random variable for all $ a, b $ in $ \mathbf{R} $. Consider the following st...

rajveer43

138 views

rajveer43 asked Jan 11

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