List of top Statistical Science Questions on Probability asked in ICAR AIEEA (PG)

If \(A\) and \(B\) are two events such that \[ P(A \cup B)=\frac{5}{6}, \qquad P(A \cap B)=\frac{1}{3}, \qquad P(B')=\frac{1}{2}, \] then the events \(A\) and \(B\) are:

If \(X\) is a random variable with probability density function \[ f(x)=\frac{1}{\sqrt{12\pi}}\,e^{-\frac{(x-1)^2}{12}}, \qquad 0<x<\infty, \] then the value of \[ \int_{1}^{\infty} f(x)\,dx \] is:

The annual precipitation data of a city is normally distributed with mean 1200 mm and standard deviation 200 mm. The probability that annual precipitation will be more than 1400 mm is

If \(X\) is a normal random variable with mean 50 and variance 25, then \(P(X \le 55)\) is

A die is thrown 100 times. Getting an even number is considered a success. The variance of the number of successes is

Two events, A and B, are mutually exclusive and each have a nonzero probability. If event A is known to occur, the probability of the occurrence of event B is:

A random variable \(X\) has the following pdf:
\[ f(x) = \frac{1}{\sqrt{12\pi}} e^{-\frac{(x-1)^2}{6}}; \quad -\infty < x < \infty \]
\(= 0\), otherwise.
Then \(\int_{1}^{\infty} f(x)dx = ?\)

The annual precipitation data of a city is normally distributed with mean \(1200\text{mm}\) and standard deviation \(200\text{mm}\), respectively. The probability that annual precipitation will be more than \(1400\text{mm}\)

A bowl contains eight chips. Three of the chips are red and the remaining five are blue. Two chips are to be drawn successively, at random and without replacement. What is the probability that the first draw results in a red chip and that the second draw results in a blue chip?

The probability of occurrence of two events \(A\) and \(B\) are 0.25 and 0.50 respectively. The probability of their simultaneous occurrence is 0.14. The probability that neither \(A\) occurs nor \(B\) occurs is:

A die is thrown 100 times. Getting an even number is considered a success. The variance of the number of successes is:

If \(X\) is a random Poisson variate with \(P(X=1) = P(X=2)\). Then \(P(X=4)\) is:

Two dice are rolled simultaneously. What is the probability that the sum of numbers on dice is 9?

Given \(P(A) = p_1, P(B) = p_2\) and \(P(AB) = p_3\). Then \(P(A' \cap B')\) will be:

If $C_1$ and $C_2$ are subset of sample space S. Then which is correct:

Two unbiased dice are thrown simultaneously. Then the possible number of outcomes will be:

In one-way ANOVA, if the calculated F value is less than the table F value then:

If a hypothesis is rejected at the 0.025 level of significance, it:

The run scored by Virat Kohli during IPL 2023 are as follows:
2, 45, 65, 7, 5, 14, 26, 46, 43, 112, 113, 9, 13, and 49. Find the median score.

In a one-tail test for the population mean, if the null hypothesis is not rejected when the alternative hypothesis is true, then:

If the mean and variance of a variable following binomial distribution are 16 and 8 respectively, then find $P(X > 0)$.

Let X be a random variable with p.d.f. f(x) =6x(1-x), 0\(\le\) x \(\le\)1. Determine the value of b such that P(X < b)=P(X > b).
1. \(\frac{1}{2}\)
2. \(\frac{1 \mp \sqrt{3}}{2}\)
3. 0
4. \(\frac{1}{3}\)

If X and Y are independent Poisson variates, What is the distribution of \(\frac{X}{X+Y}\)?
1. Binomial distribution
2. Poisson distribution
3. Negative binomial distribution
4. Multinomial distribution

The range of Correlation coefficient is
1. -1 to +1
2. -1 to 0
3. 0 to +1
4. -\(\infty\) to +\(\infty\)

The relationship between root mean square deviation (r.m.s.d.) and standard deviation (s.d.) is
1. s.d. = highest value of r.m.s.d.
2. s.d. = least value of r.m.s.d.
3. s.d. = r.m.s.d.
4. s.d. = (r.m.s.d)/2