List of top Statistics Questions on Probability asked in Indian Institute Of Technology Joint Admission Test for MSc

Two factories \( F_1 \) and \( F_2 \) produce cricket bats that are labelled. Any randomly chosen bat produced by factory \( F_1 \) is defective with probability 0.5 and any randomly chosen bat produced by factory \( F_2 \) is defective with probability 0.1. One of the factories is chosen at random, and two bats are randomly purchased from the chosen factory. Let the labels on these purchased bats be \( B_1 \) and \( B_2 \). If \( B_1 \) is found to be defective, then the conditional probability that \( B_2 \) is also defective is equal to __________ (round off to 2 decimal places).

If 12 fair dice are independently rolled, then the probability of obtaining at least two sixes is equal to __________ (round off to 2 decimal places)

Consider a coin for which the probability of obtaining head in a single toss is \( \frac{1}{3} \). Sunita tosses the coin once. If head appears, she receives a random amount of \( X \) rupees, where \( X \) has the \( \text{Exp}\left( \frac{1}{9} \right) \) distribution. If tail appears, she loses a random amount of \( Y \) rupees, where \( Y \) has the \( \text{Exp}\left( \frac{1}{3} \right) \) distribution. Her expected gain (in rupees) is equal to __________ (answer in integer).

Consider four dice \( D_1, D_2, D_3, \) and \( D_4 \), each having six faces marked as follows:

Die	Marks on faces
\(D_1\)	4, 4, 4, 4, 0, 0
\(D_2\)	3, 3, 3, 3, 3, 3
\(D_3\)	6, 6, 2, 2, 2, 2
\(D_4\)	5, 5, 5, 1, 1, 1

In each roll of a die, each of its six faces is equally likely to occur. Suppose that each of these four dice is rolled once, and the marks on their upper faces are noted. Let the four rolls be independent. If \( X_i \) denotes the mark on the upper face of the die \( D_i \), \( i = 1, 2, 3, 4 \), then which of the following statements is/are correct?

Let \( X \) and \( Y \) be independent random variables having \( \text{Bin}(18, 0.5) \) and \( \text{Bin}(20, 0.5) \) distributions, respectively. Further, let \( U = \min\{X, Y\} \) and \( V = \max\{X, Y\} \). Then which of the following statements is/are correct?

Let the random vector \( (X, Y) \) have the joint probability density function
\[f(x, y) = \begin{cases} \frac{1}{x}, & \text{if } 0 < y < x < 1 \\0, & \text{otherwise} \end{cases}.\]
Then \( \text{Cov}(X, Y) \) is equal to:

Two fair coins \( S_1 \) and \( S_2 \) are tossed independently once. Let the events \( E, F, \) and \( G \) be defined as follows:
\( E \): Head appears on \( S_1 \)
\( F \): Head appears on \( S_2 \)  
\( G \): The same outcome (head or tail) appears on both \( S_1 \) and \( S_2 \)  
Then which of the following statements is NOT correct?

Let \( Y \) be a continuous random variable such that \( P(Y > 0) = 1 \) and \( \mathbb{E}(Y) = 1 \). For \( p \in (0, 1) \), let \( \xi_p \) denote the \( p \)th quantile of the probability distribution of the random variable \( Y \). Then which of the following statements is always correct?

A bag has 5 blue balls and 15 red balls. Three balls are drawn at random from the bag simultaneously. Then the probability that none of the chosen balls is blue equals

For \( n \in \mathbb{N} \), let \( Z_n \) be the smallest order statistic based on a random sample of size \( n \) from the \( U(0,1) \) distribution. Let \( nZ_n \xrightarrow{d} Z \) as \( n \to \infty \) for some random variable \( Z \). Then \( P(Z \leq \ln 3) \) is equal to:

Let 𝑋₁~Gamma(1,4), 𝑋₂~Gamma(2, 2) and 𝑋₃~Gamma(3, 4) be three independent random variables. If 𝑌=𝑋₁+2𝑋₂+𝑋₃, then 𝐸((\(\frac{y}{4}\))⁴ ) equals ______

Let 𝑋₁~𝑁(2, 1), 𝑋₂~𝑁(−1,4) and 𝑋₃~𝑁(0, 1) be mutually independent random variables. Then, the probability that exactly two of these three random variables are less than 1, equals____(round off to two decimal places)

A box contains a certain number of balls out of which 80% are white, 15% are blue and 5% are red. All the balls of the same color are indistinguishable. Among all the white balls, 𝛼% are marked defective, among all the blue balls, 6% are marked defective and among all the red balls, 9% are marked defective. A ball is chosen at random from the box. If the conditional probability that the chosen ball is white, given that it is defective, is 0.4, then 𝛼 equals ________

Let 𝑋₁, 𝑋₂, 𝑋₃ be 𝑖. 𝑖. 𝑑. random variables, each having the 𝑁(0, 1) distribution. Then, which of the following statements is/are TRUE?

Let 𝑋 and 𝑌 be i.i.d. random variables each having the 𝑁(0, 1) distribution. Let 𝑈=\(\frac{𝑋}{𝑌}\) and 𝑍=|𝑈|. Then, which of the following statements is/are TRUE?

Let 𝑋 be a random variable having a probability mass function 𝑝(𝑥) which is positive only for non-negative integers. If
𝑝(𝑥+1)=(\(\frac{ ln\,\, 3}{𝑥+1}\)) 𝑝(𝑥), 𝑥=0, 1, 2, … , 
then 𝑉𝑎𝑟(𝑋) equals

Let 𝑋₁,𝑋₂, … be a sequence of independent random variables such that
\(p(x_i=1)=\frac{1}{4}\) and \(p(x_i=21)=\frac{3}{4}\) i=1,2,….
For some real constants 𝑐₁ and 𝑐₂, suppose that

Then, the value of \(\sqrt{3}\) (3𝑐₁+𝑐₂) equals

Let 𝐸₁, 𝐸₂ and 𝐸₃ be three events such that 𝑃(𝐸₁∩𝐸₂ )=\(\frac{1}{4}\) , 𝑃(𝐸₁ ∩ 𝐸₃ )=𝑃(𝐸₂ ∩ 𝐸₃ )=\(\frac{1}{5}\) and 𝑃(𝐸₁ ∩ 𝐸₂ ∩ 𝐸₃ ) =\(\frac{1}{6}\) . Then, among the events 𝐸₁ 𝐸₂ and 𝐸₃, the probability that at least two events occur, equals

Let 𝑌~𝐹_4,2 . Then, 𝑃(𝑌≤2) equals

Let 𝑋~𝐹_6,2 and 𝑌~𝐹_2,6 . If 𝑃(𝑋 ≤ 2) =\( \frac{216}{343}\) and 𝑃 (𝑌≤\(\frac{1}{2}\) )=𝛼, then 686𝛼 equals

If M(t), t ∈ \(\R\), is the moment generating function of a random variable, then which one of the following is NOT the moment generating function of any random variable ?

Consider a sequence of independent Bernoulli trials, where \(\frac{3}{4}\) is the probability of success in each trial. Let X be a random variable defined as follows : If the first trial is a success, then X counts the number of failures before the next success. If the first trial is a failure, then X counts the number of successes before the next failure. Then 2E(X) equals __________

Let X and Y be two independent and identically distributed random variables having U(0, 1) distribution. Then P(X² < Y < X) equals __________ (round off to 2 decimal places)

Let Ω = {1, 2, 3, … } be the sample space of a random experiment and suppose that all subsets of Ω are events. Further, let P be a probability function such that P({i}) > 0 for all i ∈ Ω. Then which of the following statements is/are true ?

Suppose that a fair coin is tossed repeatedly and independently. Let X denote the number of tosses required to obtain for the first time a tail that is immediately preceded by a head. Then E(X) and P(X > 4), respectively, are