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A die is tossed once. If the random variable X is defined as $X = \begin{cases} 1, & \text{if the die result in an odd number} \\ -1, & \text{if the die result in an even number} \end{cases}$, then the variance of X is

'A' speaks the truth in 80% of the cases while 'B' in 90% of the cases. The probability that they contradict each other in stating the same statement is

Match List-I with List-II

Let $A$ and $B$ be two events such that $P(A) = 0.2$, $P(B) = 0.4$, $P(B|A) = 0.5$

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A random variable 'X' denotes the number of sixes obtained in three throws of a die. Then, the mean of the distribution is:-

A fair coin is tossed 100 times. The probability of getting head an odd number of times is

A bag contains 6 red balls, 4 green balls and 10 blue balls. Three balls are drawn with replacement. The probability of getting at least 1 green ball is:

A random variable y has the following probability distribution

Match List-I with List-II

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Let x denotes the number of heads in a simultaneous toss of three coins, then $P(0 < x \leq 3)$

The probabilities of occurrance of two events A and B are 0.45 and 0.20 respectively. The probability of their simultaneous occurrence is 0.06. The probability that neither A nor B occurs is

Let A and B be two events. Then which of the following statements are TRUE?

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

(B) $P(B') = 1 + P(B)$

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

(D) $P(A \cap B) = P(A).P(B)$ If A and B are independent events

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Match List-I with List-II

If the random variable x has the following distribution:

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A bag contain 8 blue and 12 green balls. Two balls are drawn in succession without replacement. The probability that first is blue and second is green is

If the binomial distribution $X\sim B(n, p)$ of mean 3 and variance $\frac{3}{2}$, $(p + q) = 1$, then which of the following is/are TRUE?

(A) $q = \frac{1}{2}$, $n = 6$

(B) $P(X \leq 5) = \frac{63}{64}$, $p = \frac{1}{2}$

(C) $q = \frac{1}{3}$, $p = \frac{2}{3}$

(D) $P(X = 4) = \frac{15}{64}$, $n = 6$

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A random variable X follow Poisson distribution such that P(X=1) = 2P(X=2) , then P(X=0) is

A dice is thrown twice, the probability of occurence of 5 at least once is

Let X be random variable which assumes $x_1$, $x_2$, $x_3$, $x_4$ such that 2P(X=$x_1$)= 3P(X=$x_2$)=P(X=$x_3$)=5P(X=$x_4$) , then the probability distribution of X is

Two cards are drawn simultaneously at random from a well shuffled pack of 52 Cards. Let X be the random variable which denotes number of kings in the draw. Then the probability distribution of X is

A coin is tossed and a die is thrown. The probability that the outcome will be a tail on the coin or a number greater than 3 on the die is

If A and B are independent events, then which of the following is/are true?

(A) $\bar{A}$ and B are independent events

(B) $P(A \cap B) = 0$

(C) $\bar{A}$ and $\bar{B}$ are independent events

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

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If $P(A) = \frac{3}{5}$, $P(B) = \frac{1}{2}$ and $P(A \cap B) = \frac{1}{4}$, then $P(\overline{A} | \overline{B})$ is

Let box I contains 3 black and 4 white balls, box II contains 2 black and 2 white balls, box III contains 4 black and 3 white balls. A box is selected at random and then a ball is randomly drawn from the selected box. If the color of the ball is black then the probability that the ball is drawn from box III, is:

Let X be a random variable. Let E (X) and Var (X) denote the mean and the variance of X respectively. Then match List-I with List-II

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It is given that 3% of items manufactured by an industry are defective. The probability that a packet of 250 items contains one defective item is: [Given: $e^{-7.5} \approx 0.000553$]

For independent events $A_1, A_2, A_3, ..., A_n$ if $P(A_i) = \frac{1}{i+1}$, $i = 1, 2, 3, ..., n$, then the probability that none of the events occur is:

A fair coin is tossed a fixed number of times. If the probability of getting 11 heads is equal to the probability of getting 13 heads, then the probability of getting 2 heads is:

If the random variable X has the following probability distribution:

Match List-I with List-II

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1. (A) - (II), (B) - (I), (C) - (IV), (D) - (III)
2. (A) - (IV), (B) - (I), (C) - (II), (D) - (III)
3. (A) - (IV), (B) - (II), (C) - (I), (D) - (III)
4. (A) - (III), (B) - (II), (C) - (I), (D) - (IV)

If the events A and B are independent, then which of the following statements are true?

(A) P(A'B) = [1-P(A)] P(B)

(B) A and B are mutually exclusive

(C) P(A) = P(B)

(D) P(A'B') = [1-P(A)] [1-P(B)]

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If A and B are two distinct events such that P(A|B) = P(B|A), then which of the following is /are possible?

(A) A= B

(B) P (A) = P(B)

(C) A ⊂ B but A ≠ B

(D) A∩ B = ɸ

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A and B throw a die alternatively till one of them gets 3 or 6 and wins the game. If B starts the game, then the probability of winning the game by A is

Let A, B, C be three events. If the probability of occurring exactly one out of A and B is $\frac{3}{5}$, exactly one of B and C is $\frac{1}{5}$, exactly one of C and A is $\frac{3}{5}$ and that of occurring of three events is $\frac{4}{25}$, then the probability of occurring at least one of them is