Recent questions and answers in Probability - GATE CSE Doubts

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GATE GURU TEST SERIES MATHS
A cancer institute has a testing methodology. The methodology produces 95% positive result, also 10% rate of false-positive(i..e showing a non cancer person as a cancer affected patient). If there are 0.5% of the people have cancer. Suppose a person is classified as a disease affected person, what is the probability that the person is really have cancer? _______________

answered 3 days ago in Probability by shaktipratap (9 points) | 95 views

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A first course in probability by Ross
Let A denote the event that the midtown temperature in Los Angeles is 70◦F, and let B denote the event that the midtown temperature in New York is 70◦F. Also, let C denote the event that the maximum of the midtown temperatures in New York and in Los ... ,P(B) =.4, and P(C) = .2, find the probability that the minimum of the two midtown temperatures is 70◦F.

asked May 21 in Probability by Neelam_$ingh_222 (11 points) | 3 views

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#A First Course In Probability Sheldon Ross #self_doubt

asked May 21 in Probability by Neelam_$ingh_222 (11 points) | 5 views

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A first course in probability by Ross
A basketball team consists of 6 frontcourt and 4 backcourt players. If players are divided into roommates at random, what is the probability that there will be exactly two roommate pairs made up of a backcourt and a frontcourt player?

asked May 21 in Probability by Neelam_$ingh_222 (11 points) | 3 views

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A first course in probability
Let A denote the event that the midtown temperature in Los Angeles is 70◦F, and let B denote the event that the midtown temperature in New York is 70◦F. Also, let C denote the event that the maximum of the midtown temperatures in New York and in Los Angeles is ... ,P(B) = .4, and P(C) = .2, find the probability that the minimum of the two midtown temperatures is 70◦F.

asked May 21 in Probability by Neelam_$ingh_222 (11 points) | 2 views

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GATE2020-CS-17 Video Solution
Let $\mathcal{R}$ be the set of all binary relations on the set $\{1,2,3\}$. Suppose a relation is chosen from $\mathcal{R}$ at random. The probability that the chosen relation is reflexive (round off to $3$ decimal places) is ______.

answered Apr 23 in Probability by amitkhurana512 (187 points) | 3 views

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GATE2012-33 Video Solution
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 $6$ ? $\dfrac{10}{21}$ $\dfrac{5}{12}$ $\dfrac{2}{3}$ $\dfrac{1}{6}$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2017-1-19 Video Solution
Let $X$ be a Gaussian random variable with mean 0 and variance $\sigma ^{2}$. Let $Y$ = $\max\left ( X,0 \right )$ where $\max\left ( a,b \right )$ is the maximum of $a$ and $b$. The median of $Y$ is ______________ .

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2014-1-2 Video Solution
Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2005-IT-32 Video Solution
An unbiased coin is tossed repeatedly until the outcome of two successive tosses is the same. Assuming that the trials are independent, the expected number of tosses is $3$ $4$ $5$ $6$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE1994-1.4, ISRO2017-2 Video Solution
Let $A$ and $B$ be any two arbitrary events, then, which one of the following is TRUE? $P (A \cap B) = P(A)P(B)$ $P (A \cup B) = P(A)+P(B)$ $P (A \mid B) = P(A \cap B)P(B)$ $P (A \cup B) \leq P(A) + P(B)$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2013-24 Video Solution
Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is $\dfrac{1}{2}.$ What is the expected number of unordered cycles of length three? $\dfrac {1}{8}$ $1$ $7$ $8$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2010-27 Video Solution
What is the probability that divisor of $10^{99}$ is a multiple of $10^{96}$? $\left(\dfrac{1}{625}\right)$ $\left(\dfrac{4}{625}\right)$ $\left(\dfrac{12}{625}\right)$ $\left(\dfrac{16}{625}\right)$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2017-2-31 Video Solution
For any discrete random variable $X$, with probability mass function $P(X=j)=p_j, p_j \geq 0, j \in \{0, \dots , N \}$, and $\Sigma_{j=0}^N \: p_j =1$, define the polynomial function $g_x(z) = \Sigma_{j=0}^N \: p_j \: z^j$. For a certain discrete ... . The expectation of $Y$ is $N \beta(1-\beta)$ $N \beta$ $N (1-\beta)$ Not expressible in terms of $N$ and $\beta$ alone

asked Apr 18 in Probability by admin (3.6k points) | 7 views

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GATE2017-2-48 Video Solution
If a random variable $X$ has a Poisson distribution with mean $5$, then the expectation $E\left [ \left ( x+2 \right )^{2} \right ]$ equals ___.

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2008-29 Video Solution
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 $P (X ≤ -1) = P (Y ≥ 2)$ , the standard deviation of $Y$ is $3$ $2$ $\sqrt{2}$ $1$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2007-24 Video Solution
Suppose we uniformly and randomly select a permutation from the $20 !$ permutations of $1, 2, 3\ldots ,20.$ What is the probability that $2$ appears at an earlier position than any other even number in the selected permutation? $\left(\dfrac{1}{2} \right)$ $\left(\dfrac{1}{10}\right)$ $\left(\dfrac{9!}{20!}\right)$ None of these

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2018-44 Video Solution
Consider Guwahati, (G) and Delhi (D) whose temperatures can be classified as high $(H)$, medium $(M)$ and low $(L)$. Let $P(H_G)$ denote the probability that Guwahati has high temperature. Similarly, $P(M_G)$ and $P(L_G)$ ... , then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is _____

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2019-47 Video Solution
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) _______

asked Apr 18 in Probability by admin (3.6k points) | 2 views

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GATE2001-2.4 Video Solution
Seven (distinct) car accidents occurred in a week. What is the probability that they all occurred on the same day? $\dfrac{1}{7^7}\\$ $\dfrac{1}{7^6}\\$ $\dfrac{1}{2^7}\\$ $\dfrac{7}{2^7}\\$

asked Apr 18 in Probability by admin (3.6k points) | 0 views

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GATE2017-2-26 Video Solution
$P$ and $Q$ are considering to apply for a job. The probability that $P$ applies for the job is $\dfrac{1}{4},$ the probability that $P$ applies for the job given that $Q$ applies for the job is $\dfrac{1}{2},$ and the probability that $Q$ applies for the job ... $\left(\dfrac{5}{6}\right)$ $\left(\dfrac{7}{8}\right)$ $\left(\dfrac{11}{12}\right)$

asked Apr 18 in Probability by admin (3.6k points) | 2 views

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GATE2009-21 Video Solution
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 probability ... following options is closest to the probability that the face value exceeds $3$? $0.453$ $0.468$ $0.485$ $0.492$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2016-1-29 Video Solution
Consider the following experiment. Step 1. Flip a fair coin twice. Step 2. If the outcomes are (TAILS, HEADS) then output $Y$ and stop. Step 3. If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output $N$ and stop. Step 4. If the ... (TAILS, TAILS), then go to Step 1. The probability that the output of the experiment is $Y$ is (up to two decimal places)

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2004-IT-33 Video Solution
Let $X$ and $Y$ be two exponentially distributed and independent random variables with mean $α$ and $β$, respectively. If $Z$ = min $(X, Y)$, then the mean of $Z$ is given by $\left(\dfrac{1}{\alpha + \beta}\right)$ $\min (\alpha, \beta)$ $\left(\dfrac{\alpha\beta}{\alpha + \beta}\right)$ $\alpha + \beta$

asked Apr 18 in Probability by admin (3.6k points) | 2 views

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GATE2016-1-04 Video Solution
A probability density function on the interval $[a, 1]$ is given by $1/x^{2}$ and outside this interval the value of the function is zero. The value of $a$ is _________.

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2011-34 Video Solution
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 the probability that the two cards are selected with the number on the first card being one higher than the number ... $\left(\dfrac{4}{25}\right)$ $\left(\dfrac{1}{4}\right)$ $\left(\dfrac{2}{5}\right)$

asked Apr 18 in Probability by admin (3.6k points) | 2 views

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GATE2018-15 Video Solution
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 there is no tie. Define a ... and that all trials are independent. The probability (rounded to $3$ decimal places) that one of them wins on the third trial is ____

asked Apr 18 in Probability by admin (3.6k points) | 7 views

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GATE2012-21 Video Solution
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 $0$ and $0.5$ $0$ and $1$ $0.5$ and $1$ $0.25$ and $0.75$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2015-3-37 Video Solution
Suppose $X_i$ for $i=1, 2, 3$ are independent and identically distributed random variables whose probability mass functions are $Pr[X_i = 0] = Pr[X_i = 1] = \frac{1} {2} \text{ for } i = 1, 2, 3$. Define another random variable $Y = X_1X_2 \oplus X_3$, where $\oplus$ denotes XOR. Then $Pr[Y=0 \mid X_3 = 0] =$______.

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE1995-1.18 Video Solution
The probability that a number selected at random between $100$ and $999$ (both inclusive) will not contain the digit $7$ is: $\dfrac{16}{25}$ $\left(\dfrac{9}{10}\right)^{3}$ $\dfrac{27}{75}$ $\dfrac{18}{25}$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2003-60, ISRO2007-45 Video Solution
A program consists of two modules executed sequentially. Let $f_1(t)$ and $f_2(t)$ ... $\int_0^t f_1(x)f_2(t-x)dx$ $\max\{f_1(t),f_2(t)\}$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2013-2 Video Solution
Suppose $p$ is the number of cars per minute passing through a certain road junction between $5$ PM and $6$ PM, and $p$ has a Poisson distribution with mean $3$. What is the probability of observing fewer than $3$ cars during any given minute in this interval? $\dfrac{8}{(2e^{3})}$ $\dfrac{9}{(2e^{3})}$ $\dfrac{17}{(2e^{3})}$ $\dfrac{26}{(2e^{3})}$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2004-80 Video Solution
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 $p^{2}$ is $\left(\dfrac{2}{3}\right)$ $\quad 1$ $\left(\dfrac{4}{3}\right)$ $\left(\dfrac{5}{3}\right)$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2007-IT-57 Video Solution
In a multi-user operating system on an average, $20$ requests are made to use a particular resource per hour. The arrival of requests follows a Poisson distribution. The probability that either one, three or five requests are made in $45$ ... $6.9 \times 10^3 \times e^{-20}$ $1.02 \times 10^3 \times e^{-20}$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE1995-2.14 Video Solution
A bag contains $10$ white balls and $15$ black balls. Two balls are drawn in succession. The probability that one of them is black and the other is white is: $\frac{2}{3}$ $\frac{4}{5}$ $\frac{1}{2}$ $\frac{1}{3}$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2016-2-05 Video Solution
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 _________.

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2002-2.16 Video Solution
Four fair coins are tossed simultaneously. The probability that at least one head and one tail turn up is $\frac{1}{16}$ $\frac{1}{8}$ $\frac{7}{8}$ $\frac{15}{16}$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2014-1-48 Video Solution
Four fair six-sided dice are rolled. The probability that the sum of the results being $22$ is $\dfrac{X}{1296}$. The value of $X$ is _______

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2005-12, ISRO2009-64 Video Solution
Let $f(x)$ be the continuous probability density function of a random variable $x$, the probability that $a < x \leq b$, is : $f(b-a)$ $f(b) - f(a)$ $\int\limits_a^b f(x) dx$ $\int\limits_a^b xf (x)dx$

asked Apr 18 in Probability by admin (3.6k points) | 1 view

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GATE2003-3 Video Solution
Let $P(E)$ denote the probability of the event $E$. Given $P(A) = 1$, $P(B) =\dfrac{1}{2}$, the values of $P(A\mid B)$ and $P(B\mid A)$ respectively are $\left(\dfrac{1}{4}\right),\left(\dfrac{1}{2}\right)$ $\left(\dfrac{1}{2}\right),\left(\dfrac{1}{4}\right)$ $\left(\dfrac{1}{2}\right),{1}$ ${1},\left(\dfrac{1}{2}\right)$

asked Apr 18 in Probability by admin (3.6k points) | 2 views

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