Probability and Statistics
Practice QuestionsMarks 1
1
P and Q play chess frequently against each other. Of these matches, P has won $80 \%$ of the matches, drawn $15 \%$ of the matches and lost $5 \%$ of the matches. If they play 3 more matches, what is the probability of $P$ winning exactly 2 of these 3 matches?
GATE ME 2025
2
If two unbiased coins are tossed, then what is the probability of having at least one head?
GATE ME 2025
3
A machine produces a defective component with a probability of 0.015. The number of defective components in a packed box containing 200 components produced by the machine follows a Poisson distribution. The mean and the variance of the distribution are
GATE ME 2023
4
Two coins are tossed simultaneously. The probability (upto two decimal points accuracy) of getting at least one head is _________.
GATE ME 2017 Set 2
5
A simple of $$15$$ data is as follows; $$17,18,17,17,13,18,5,5,6,7,8,9,20,17,3.$$ The mode of the data is
GATE ME 2017 Set 2
6
A six-face fair dice is rolled a large number of times. The mean value of the outcomes is _________.
GATE ME 2017 Set 1
7
Consider a Poisson distribution for the tossing of a biased coin. The mean for this distribution is $$\mu $$.
The standard deviation for this distribution is given by
GATE ME 2016 Set 1
8
The area (in percentage) under standard normal distribution curve of variable $$Z$$ within limits from $$-3$$ to $$+3$$ is ________.
GATE ME 2016 Set 3
9
The probability of obtaining at least two $$'SIX'$$ in throwing a fair dice $$4$$ times is
GATE ME 2015 Set 1
10
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 ME 2015 Set 1
11
Among the four normal distributions with probability density functions as shown below, which one has the lowest variance?
GATE ME 2015 Set 1
12
If $$P(X)$$ $$ = $$ $$1/4,$$ $$P(Y) = 1/3,$$ and $$\,\,P\left( {X \cap Y} \right) = 1/12,\,\,$$ the value of $$P(Y/X)$$ is
GATE ME 2015 Set 3
13
A nationalized bank has found that the daily balance available in its savings accounts follows a normal distribution with a mean of Rs. $$500$$ and a standard deviation of Rs. $$50.$$ The percentage of savings account holders, who maintain an average daily balance more than Rs. $$500$$ is
GATE ME 2014 Set 4
14
The number of accidents occurring in a plant in a month follows Poisson distribution with mean as $$5.2.$$ The probability of occurrence of less than $$2$$ accidents in the plant during a randomly selected month is _________.
GATE ME 2014 Set 4
15
A group consists of equal number of men and women. Of this group $$20$$% of the men and $$50$$% of the women are unemployed. If a person is selected at random from this group, the probability of the selected person being employed is
GATE ME 2014 Set 3
16
Let $$X$$ be a normal random variable with mean $$1$$ and variance $$4.$$ The probability $$P\left\{ {X < 0} \right\}$$ is
GATE ME 2013
17
A box contains $$2$$ washers, $$3$$ nuts and $$4$$ bolts. Items are drawn from the box at random one at a time without replacement. The probability of drawing $$2$$ washers first followed by $$3$$ nuts and subsequently the $$4$$ bolts is
GATE ME 2010
18
If three coins are tossed simultaneously, the probability of getting at least one head is
GATE ME 2009
19
The standard deviation of a uniformly distributed random variable $$b/w$$ $$0$$ and $$1$$ is
GATE ME 2009
20
A coin is tossed $$4$$ times. What is the probability of getting heads exactly $$3$$ times?
GATE ME 2008
21
A lot had $$10$$% defective items. Ten items are chosen randomly from this lot. The probability that exactly $$2$$ of the chosen items are defective is
GATE ME 2005
22
Using given data points tabulated below, a straight line passing through the origin is fitted using least squares method. The slope of the line is
GATE ME 2005
23
A regression model is used to express a variable $$Y$$ as a function of another variable $$X.$$ This implies that
GATE ME 2002
24
The probability that two friends share the same birth-month is
GATE ME 1998
Marks 2
1
Let $X$ be a continuous random variable defined on $[0, 1]$ such that its probability density function $f(x) = 1$ for $0 \leq x \leq 1$ and $0$ otherwise. Let $Y = \log_e (X + 1)$. Then the expected value of $Y$ is _____ . (rounded off to 2 decimal places)
GATE ME 2024
2
Let a random variable X follow Poisson distribution such that
Prob(X = 1) = Prob(X = 2).
The value of Prob(X = 3) is __________ (round off to 2 decimal places).
GATE ME 2022 Set 1
3
Three cards were drawn from a pack of $$52$$ cards. The probability that they are a king, a queen, and a jack is
GATE ME 2016 Set 3
4
The probability that a screw manufactured by a company is defective is $$0.1$$. The company sells screws in packets containing $$5$$ screws and gives a guarantee of replacement if one or more screws in the packet are found to be defective. The probability that a packet would have to be replaced is _________.
GATE ME 2016 Set 2
5
The chance of a student passing an exam is $$20$$% The chance of a student passing the exam and getting above $$90$$% marks in it is $$5$$%. Given that a student passes the examination, the probability that the student gets above $$90$$% marks is
GATE ME 2015 Set 2
6
A coin is tossed thrice. Let $$X$$ be the event that head occurs in each of the first two tosses. Let $$Y$$ be the event that a tail occurs on the third toss. Let $$Z$$ be the event that two tails occur in three tosses. Based on the above information, which one of the following statements is TRUE?
GATE ME 2015 Set 3
7
In the following table, $$x$$ is a discrete random variable and $$p(x)$$ is the probability density. The standard deviation of $$x$$ is
GATE ME 2014 Set 1
8
A box contains $$25$$ parts of which $$10$$ are defective. Two parts are being drawn simultaneously in a random manner from the box. The probability of both the parts being good is
GATE ME 2014 Set 2
9
Consider an unbiased cubic die with opposite faces coloured identically and each face coloured red, blue or green such that each color appears only two times on the die. If the die is thrown thrice, the probability of obtaining red colour on top face of the die at least twice is ________.
GATE ME 2014 Set 2
10
A machine produces $$0, 1$$ or $$2$$ defective pieces in a day with associated probability of $${1 \over 6},{2 \over 3}$$ and $${1 \over 6}$$, respectively. Then mean value and the variance of the number of defective pieces produced by
GATE ME 2014 Set 3
11
The probability that a student knows the correct answer to a multiple choice question is $${2 \over 3}$$. If the student does not know the answer, then the student guesses the answer. The probability of the guessed answer being correct is $${1 \over 4}$$. Given that the student has answered the question correctly, the conditional probability that the student knows the correct answer is
GATE ME 2013
12
A box contains $$4$$ red balls and $$6$$ black balls. Three balls are selected randomly from the box one after another, without replacement. The probability that the selected set contains one red ball and two black balls is
GATE ME 2012
13
An automobile plant contracted to buy shock absorbers from two suppliers $$X$$ and $$Y$$. $$X$$ supplies $$60$$% and $$Y$$ supplies $$40$$% of the shock absorbers. All shock absorbers are subjected to a quality test. The ones that pass the quality test are considered reliable. Of $$X'$$ s shock absorbers, $$96$$% are reliable. Of $$Y'$$ s shock absorbers, $$72$$% are reliable. The probability that a randomly choosen shock absorber, which is found to reliable, is made by $$Y$$ is
GATE ME 2012
14
An unbiased coin is tossed five times. The outcome of each loss is either a head or a tail. Probability of getting at least one head is _______.
GATE ME 2011
15
Let $$X$$ and $$Y$$ be two independent random variables. Which one of the relations $$b/w$$ expectation $$(E),$$ variance $$\left( {{V_{ar}}} \right)$$ and covariance $$\left( {{C_{ov}}} \right)$$ given below is FALSE?
GATE ME 2007
16
A single die is thrown two times. What is the probability that the sum is neither $$8$$ nor $$9?$$
GATE ME 2005
17
From a pack of regular playing cards, two cards are drawn at random. What is the probability that both cards will be kings, if the card is NOT replaced?
GATE ME 2004
18
The following data about the flow of liquid was observed in a continuous chemical process plant
Mean flow rate of the liquid is
GATE ME 2004
19
In a manufacturing plant, the probability of making a defective bolts is $$0.1$$. The mean and standard deviation of defective bolts in a total of $$900$$ bolts are respectively
GATE ME 2000
20
Four arbitrary points $$\,\,\,\left( {{x_1},{y_1}} \right),\,\,\left( {{x_2},{y_2}} \right),\,\,\left( {{x_3},{y_3}} \right),\,\,\left( {{x_4},{y_4}} \right),\,\,\,\,$$ are given in the $$xy$$ $$-$$ plane using the method of least squares. If regression of $$y$$ upon $$x$$ gives the fitted line $$y=ax+b;$$ and regression of $$x$$ upon $$y$$ gives the fitted line $$x=cy+d,$$ then
GATE ME 1999