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The curve $y = f(x)$ is normal probability curve, then which of the following statements are correct?

(A) mean, median and mode of the distribution coincide.

(B) the area bounded by the curve $y = f(x)$ and $x$-axis is one unit.

(C) The curve is symmetrical about the line $x = \mu$, where $\mu$ is the mean.

(D) $y$-axis is an asymptote to the curve.

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A random variable X has the following probability distribution:

Then the value of $\frac{k}{4}$ is:

60% members of a committee favour a certain proposal and 40% members oppose the proposal. A member is selected and let the random variable X = 0 if he opposes and X = 1 if he is in favour. Then the variance of the random variable X is

If A and B are independent events, then which of the following is not true?

The probability that a leap year selected at random will have 53 Mondays is

Two persons A and B throw a die alternately till one of them gets a 'three' and wins the game. The probability of A's winning if A starts first is

If the difference between mean and variance of a Binomial distribution is 1 and the difference of their squares is 5, then the probability of success is

Let X denote the number of hours a student studies on a selected day. The probability distribution of X is given by (where k is some unknown constant)

$P(X = x_i) = \begin{cases} 0.5, & \text{if } x_i = 0, \\ kx_i, & \text{if } x_i = 1, \\ k(4 - x_i), & \text{if } x_i = 2 \text{ or } 3, \\ 0, & \text{otherwise}. \end{cases}$

Then the value of k is

Two percent of the bolts manufactured in a factory are found to be defective. Using the Poisson distribution, the probability that in a sample of 100 bolts chosen at random, exactly two will be defective, is: [Given $e^{-2}=0.135$]

Which of the following statements are correct?

(A) The mean and variance of the Poisson distribution are equal.

(B) The mean and variance of a Binomial distribution are equal.

(C) An unbiased die is thrown again and again until two sixes are obtained, then the probability of obtaining the second six in the 3rd throw is $\frac{5}{108}$.

(D) If the variance of a Poisson distribution is 2, then P(X = 2) = $2e^{-2}$

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The probability distribution of a random variable $x$ is, $P(x) = \frac{k}{2^x}, x = 0, 1, 2, 3$. Then

Match List-I with List-II

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A problem in mathematics is given to three students whose chances of solving it are 1/2, 1/3, 1/4 respectively. The probability that the problem is solved is

In a college, 30% students fail in physics, 25% fail in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in physics if she has failed in mathematics is

An urn contains 5 red and 5 black balls. A ball is drawn at random, its color is noted and is returned to the urn. Moreover, 2 additional balls of the same color are put in the urn and then a ball is drawn at random. The probability that the second drawn ball is red, is:

For any events A and B of a sample space S, which of the following statements are TRUE?

(A) $P(S | B) = 1$

(B) $P(A \cap B) = P(A) + P(B) + P(A \cup B)$

(C) $P(\bar{A} | B) = 1 - P(A | B)$

(D) $P(A | B) = \frac{P(A \cap B)}{P(B)}, P(B) \neq 0$

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The probability distribution of a random discrete variable is given

If it is known that P(X=1) is the mean of P(X=0) and P(X=2).

Then the value of r is :

A pair of dice is thrown until the sum of numbers appeared is a perfect square or a non-perfect square sum appeared five times in succession. If random variable $X$ denotes the number of non perfect square sums appeared, then $P(X > 0)$ is

If the sum and difference of squares of mean and variance of a Binomial distribution is $\frac{225}{256}$ and $\frac{63}{256}$ respectively, the $P(X \geq 2)$ is:

A and B are two independent events. The probability that both events A and B occur is $\frac{1}{6}$ and the probability that neither of them occur is $\frac{1}{3}$. If P(A) = x, P(B) = y then the value of x+y is.

For a random variable x, probability distribution P(x) is given by $P(x) = \frac{k}{6}(3-x), x = 0, 1, 2$, then

Match List-I with List-II

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If A and B are two events such that P(A) ≠ 0 and P(B | A) = 1 then

Two numbers are selected without replacement at random, one at a time from the first six positive integers. Let x denotes the larger of the two numbers.

Match List-I with List-II

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Nitin has taken the subjects mathematics, physics and chemistry. The probability of him getting grade A in these subjects are respectively 0.2, 0.3 and 0.9. Getting grades in different subjects are regarded as independent events. The probability of getting A grade by him, either in mathematics or physics, is

If A and B are two events such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$ and $P(A \cap B) = \frac{1}{4}$, then which of the following statements are true?

(A) A and B are independent events

(B) $P(A | B) = \frac{3}{4}$

(C) $P(A' | B') = \frac{5}{8}$

(D) $P(A' | B) = \frac{1}{4}$

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The probability of a shooter of hitting the target is $\frac{1}{4}$. The minimum number of fire needed so that the probability of hitting the target atleast once is greater than $\frac{7}{16}$ is:

Mean and variance of a binomial distribution are 6 and 2 respectively. The probability of 2 successes will be

Probability distribution of random variable X is

Then the value of E(X) is

A random variable X has the following probability distribution:

The variance of X will be

The probability distribution of a random variable X is given by

If k > 0, then $P(0 < X \leq 2)$ is equal to

The probability that A hits a target is $\frac{1}{5}$ and the probability that B hits it is $\frac{2}{3}$. The probability that the target will be hit if both A and B shoot at it independently is: