CUET Mathematics Algebra > Probability PYQs

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

Three students A, B and C can respectively solve 50%, 25% and 20% of the problems in a book. A particular problem is selected at random from the book. The probability that at least one of them will solve the problem is

The probabilities of occurrence of two events E and F are 0.25 and 0.50 respectively. The probability of their simultaneous occurrence is 0.14. The probability that neither E nor F occurs is

A bag contains 4 red and 6 black balls. Two balls are drawn in succession without replacement. The probability that the first is red and the second is black is

The probability distribution of a random variable X is | X | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X) | 0.2 | k | k | 2k | k | Match List-I with List-II | List-I | List-II | |---|---| | (A) value of k | (I) $\frac{16}{25}$ | | (B) $P(x \geq 2)$ | (II) $\frac{9}{25}$ | | (C) $P(x = 3)$ | (III) $\frac{4}{25}$ | | (D) $P(x < 2)$ | (IV) $\frac{8}{25}$ | Choose the correct answer from the options given below:

If X is a random variable and $a$, $b$ are real numbers, then which of the following statements are true? (A) $Var(aX+b) = a^2 Var(X)$ (B) $E(aX+b)= a E(X) + b$ (C) $E(aX+b)= a E(X) - E(b)$ (D) $Var(aX+b)= a Var(X) + b$ Choose the correct answer from the options given below:

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 1/4 | 1/2 | 1/4 | then, which of the following is correct?

The binomial distribution for which the mean is 5 and variance 4, is

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. Choose the correct answer from the options given below:

A random variable X has the following probability distribution: | X | 2 | 3 | 4 | 5 | |---|---|---|---|---| | P(X) | 5/k | 7/k | 9/k | 11/k | 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) = $2e^{-2}$ Choose the **correct** answer from the options given below:

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 | List-I | List-II | |---|---| | (A) $k$ | (I) $\frac{2}{15}$ | | (B) $P(x = 1)$ | (II) $\frac{1}{5}$ | | (C) $P(1 < x < 3)$ | (III) $\frac{8}{15}$ | | (D) $P(x \geq 2)$ | (IV) $\frac{4}{15}$ | Choose the correct answer from the options given below:

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) $P(A | B) = \frac{P(A \cap B)}{P(B)}, P(B) \neq 0$ Choose the correct answer from the options given below:

The probability distribution of a random discrete variable is given | X | -1 | 0 | 1 | 2 | 3 | |---|---|---|---|---|---| | P(X) | 0.1 | $p$ | 0.3 | $q$ | $r$ | 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 | List-I | List-II | |---|---| | (A) k is equal to | (i) $\frac{1}{2}$ | | (B) P(x = 0) | (ii) 1 | | (C) P(x < 2) | (iii) $\frac{1}{6}$ | | (D) P(1 < x ≤ 2) | (iv) $\frac{5}{6}$ | Choose the correct answer from the options given below:

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 | List-I | List-II | |---|---| | (A) P(x = 2) | (i) $\frac{4}{15}$ | | (B) P(x = 3) | (ii) $\frac{1}{15}$ | | (C) P(x = 4) | (iii) $\frac{2}{15}$ | | (D) P(x = 5) | (iv) $\frac{1}{5}$ | Choose the correct answer from the options given below:

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) $P(A' | B) = \frac{1}{4}$ Choose the correct answer from the options given below:

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 | X | -2 | -1 | 0 | 1 | 2 | |---|---|---|---|---|---| | P(X) | 2/11 | 1/11 | 4/11 | 3/11 | 1/11 | Then the value of E(X) is

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | 0.2 | 0.1 | 0.3 | 0.4 | The variance of X will be

The probability distribution of a random variable X is given by | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | k | 2k | 3k | 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:

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 $P(A) = 0.2$, $P(B) = 0.4$, $P(B|A) = 0.5$ | List-I | List-II | | --- | --- | | (A) $P(A \cap B)$ | (I) $0.5$ | | (B) $P(A\vert B)$ | (II) $0.8$ | | (C) $P(A \cup B)$ | (III) $0.25$ | | (D) $P(A')$ | (IV) $0.1$ | Choose the correct answer from the options given below:

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 | y | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| | P(y) | 2k | 3k | k | 4k | 5k | Match List-I with List-II | List-I | List-II | |---|---| | (A) $P(y > 2)$ | (I) $2/5$ | | (B) k | (II) $2/3$ | | (C) $P(y \leq 3)$ | (III) $8/15$ | | (D) $P(2 \leq y \leq 4)$ | (IV) $1/15$ | Choose the correct answer from the options given below:

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 Choose the correct answer from the options given below:

Match List-I with List-II If the random variable x has the following distribution: | x | 0 | 1 | 2 | otherwise | |---|---|---|---|-----------| | P(x) | k | k | 2k | 0 | | List-I | List-II | |---|---| | (A) k | (I) $\frac{3}{4}$ | | (B) P(x ≥ 2) | (II) $\frac{1}{4}$ | | (C) P(x ≤ 2) | (III) $\frac{1}{2}$ | | (D) P(0 < x ≤ 2) | (IV) 1 | Choose the correct answer from the options given below:

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) $P(X = 4) = \frac{15}{64}$, $n = 6$ Choose the correct answer from the options given below:

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) $P(A \cap B) = P(A) + P(B)$ Choose the correct answer from the options given below:

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 | List-I | List-II | |---|---| | (A) If Var (X) = $a$, then Var (2X + 3) is | (I) 11$a$ | | (B) If E (X) = $a$, then E (2X) is | (II) 6$a$ | | (C) If Var (X) = $a$, then Var(3X - $a$) + Var ($\sqrt{2}x + \beta$) is | (III) 4$a$ | | (D) If E (X) = $\frac{5a}{12}$, then E (12X + $a$) is | (IV) 2$a$ | Choose the correct answer from the options given below:

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: | X | 0 | 1 | 2 | otherwise | |---|---|---|---|---| | P(X) | k | 3k | 5k | 0 | Match List-I with List-II | List-I | List-II | |---|---| | (A) k | (I) $\frac{13}{9}$ | | (B) E (X) | (II) $\frac{4}{9}$ | | (C) P (X ≤ 1) | (III) $\frac{8}{9}$ | | (D) P (1 ≤ X ≤ 2) | (IV) $\frac{1}{9}$ | Choose the correct answer from the options given below: 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)] Choose the correct answer from the options given below:

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 Choose the correct answer from the options given below:

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

In a Binomial distribution, the probability of getting a success is $\frac{3}{4}$ and the variance is $\frac{3}{8}$ then the probability of no success is:

Which of the following are the properties of Normal Distribution function f(x) and Normal probability curve: (A) The probability of success remains the same in each trial and the number of trials is small in number. (B) The curve is bell-shaped and is symmetrical about the mean. (C) If set of n trials are repeated N times, then frequency f(r) of r successes is given by f(r) = N.p(r) = N$e^{-m\frac{m^r}{r!}}$, r=0,1,2,... (D) As x increases numerically, f(x) decreases rapidly and the maximum value of f(x) occurs at x=μ(mean) Choose the correct answer from the options given below:

The random variable X has the following probability distribution | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | a | a | b | b | such that E(x²) = 2E(x), then the value of b is:

The probability that in a year of the 22nd century choosen at random, there will be 53 Sundays is:

The probability distribution of a random variable x is given below. | x | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(x) | k/3 | k/2 | k/4 | k/7 | Then the value of k is

If A and B are independent events and $P(A) = \frac{1}{2}$ $P(B) = \frac{1}{3}$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) $P(A \cap B)$ | (I) $\frac{1}{2}$ | | (B) $P(\bar{A})P(B) + P(A)P(\bar{B})$ | (II) $\frac{1}{3}$ | | (C) $P(A \mid B) + P(B \mid A)$ | (III) $\frac{1}{6}$ | | (D) $P(A \cap \bar{B})$ | (IV) $\frac{5}{6}$ | Choose the correct answer from the options given below:

If $P(A) = \frac{3}{10}$, $P(B) = \frac{2}{5}$ and $P(A \cup B) = \frac{3}{5}$ then the value of $P(B|A) + P(A|B)$ is:

Bag I contains 3 black and 2 white balls. Bag II contains 2 black and 4 white balls. A bag is selected at random and then a ball is drawn from it. The probability that the ball drawn is black is:

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. The probability distribution of number of aces is given by:

Suppose X has Poisson distribution such that $3 P(X=1) = 2 P(X=2)$ then $P(X>0)$ is:

For a Binomial distribution, B(n,p), where p+q=1, the sum and product of mean and variance are 8 and 12 respectively, when the value of n is:

A die is thrown 4 times and getting 3 is considered a success. The probability of 2 successes is:

The random variable X can take values 0, 1, 2. If $P(X=0)=P(X=1)=\alpha$, and $E(X^2)=E(X)$, then which of the following are correct? (A) $E(X) = 2-3\alpha$ (B) $E(X^2) = 4+7\alpha$ (C) $\alpha = \frac{1}{2}$ (D) $\alpha = \frac{1}{5}$ Choose the **correct** answer from the options given below:

Let X denotes the number of doublets obtained in 3 throws of a pair of dice. Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) $P(X = 0)$ | (I) $\frac{1}{216}$ | | (B) $P(X = 1)$ | (II) $\frac{15}{216}$ | | (C) $P(X = 2)$ | (III) $\frac{75}{216}$ | | (D) $P(X = 3)$ | (IV) $\frac{125}{216}$ | Choose the correct answer from the options given below:

Two persons A and B throw a die alternately till one of them gets a six and wins the game. If A begins, then the probabilities of winning of A and B respectively are

Consider two independent events A and B such that $P(A) = 0.3$, $P(B) = 0.6$. Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) $P(A$ and $B)$ | (I) 0.28 | | (B) $P(A$ and not $B)$ | (II) 0.18 | | (C) $P(A$ or $B)$ | (III) 0.12 | | (D) $P$(neither A nor B) | (IV) 0.72 | Choose the correct answer from the options given below:

If E and F are independent events associated with an experiment, then which one of the following statements is correct?

Probability that a man speaks truth is $\frac{3}{4}$. He throws a die and reports that it is a six. The probability that it is actually a six is

The mean of the number of heads in the two tosses of a coin is

Five dice are thrown simultaneously. If the occurrence of an even number in a single dice is considered a success, then the probability of at most 3 successes is

If X is a random variable with probability distribution as given below: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | k | 2k | k | 3k | Then, the variance of the distribution is

A random variable X has the following probability distribution: | X | -2 | -1 | 0 | 1 | 2 | 3 | |---|---|---|---|---|---|---| | P(X) | 0.1 | 0.2 | k | 0.3 | 2k | 0.1 | then which of the following are TRUE? (A) $k=0.1$ (B) $P(X < 1) = 0.4$ (C) $P(X < 2) = 0.7$ (D) $P(0 < X < 3) = 0.5$ Choose the correct answer from the options given below:

If $X$ is a random variable which can assume values $0, 1, 2, 3$ or $4$ such that $P(X = 1) = P(X = 2)$ and $3P(X = 3) = 4P(X = 4) = P(X = 0) = \frac{1}{8}$, then $P(X > 0)$ is:

A couple has 3 children each child is equally likely to be a boy or a girl. The probability that the eldest child is a girl given that they have atleast one boy is:

A letter is known to have come from either TATAPUR or from CHAKRATA. On the envelope, only two letters 'TA' are visible consecutively. The probability that the letter has come from CHAKRATA is:

Let A and B be independent events such that P (A) = 0.3 and P (B) = 0.4, then Match List-I with List-II | List-I | List-II | | ----------------- | ---------- | | (A) $P(A \cap B)$ | (I) 0.3 | | (B) $P(A \cup B)$ | (II) 0.4 | | (C) $P(A \mid B)$ | (III) 0.12 | | (D) $P(B \mid A)$ | (IV) 0.58 | Choose the correct answer from the options given below:

The probability that it will rain on any particular day is 50%. The probability that it rains only on the first 4 days of the week is:

It is known that 3% of plastic bags manufactured in a factory are defective. Using the Poisson distribution on a sample of 100 bags, the probability of at most one defective bag is:

A die is rolled in such a way that an even number is twice likely to occur as an odd number. If the die is rolled twice, then the mean of the number of perfect squares in two tosses is:

The probability distribution function of a normal variate with mean $\mu$ and variance $\sigma^2$ is given by: $f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$, $-\infty < x < \infty$, $-\infty < \mu < \infty$, $\sigma > 0$ If $y = f(x)$ be the normal probability curve, then which of the following is correct? (A) The normal curve is symmetrical about the line $x = \mu$. (B) Mean, median and mode of the distribution coincide. (C) Y- axis is an asymptote to the normal curve. (D) If x increases numerically, $f(x)$ decreases rapidly. Choose the correct answer from the options given below:

If two dice are rolled 12 times and getting a total greater than 4 is considered as a success, then which of the following statements are correct? (A) The probability of getting a total greater than 4 in a single throw of the pair of dice is 5/6. (B) Mean = 10 (C) Variance = 3/5 (D) The probability of getting a total less than or equal to 4 in a single throw of the pair of dice is 1/6. Choose the correct answer from the options given below:

If the random variable X has the following probability distribution: | X | 0 | 1 | 2 | Otherwise | |---|---|---|---|---| | P(X) | K | 3K | 5K | 0 | then K is equal to

A can solve 90% problems and B can solve 70% problems of the book. A problem is selected at random from the book. The probability that the problem is solved, is equal to

If A and B are two independent events, then which of the following is/are true? (A) $P(A \cap B) = 0$ (B) $P(A \cup B) = 1 - P(A')P(B')$ (C) $P(A \cup B) = P(A)P(B)$ (D) $P(A \cap B) = P(A)P(B)$ Choose the correct answer from the options given below:

A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be heart then the probability of the missing card to be a heart is:

Let E and F be two events such that $P(E) = \frac{1}{3}$, $P(F) = \frac{1}{4}$ and $P(E \cap F) = \frac{1}{5}$. Then the value of $P(F|E)$ is equal to

There are 50 telephone lines in an exchange. The probability that any one of them will be busy is 0.1, then the probability that all the lines are busy?

If a random variable y follows Poisson's distribution such that $P(X = 2) = 9 P(X = 4) + 90 P(X = 6)$, then sum of the mean and variance of X is

The mean of the number of heads in a simultaneous toss of three coins is

A die is thrown 6 times. If getting an odd number is a success, then the probability of at least 5 successes is

Let X denotes the number of hours a person uses a mobile and the probability distribution of X is as | X | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X) | 0.1 | K | 2K | 2K | K | Then the value of K is

A letter is known to have come either from KOLKATA or TATANAGAR. On the envelope just two consecutive letters TA are visible. The probability that letter has come from TATANAGAR is

Bag A contains 2 unbiased and 3 biased coins whereas Bag B contains 3 unbiased and 2 biased coins. A bag is selected at random and 2 coins are taken out simultaneously. The probability, that both coins are unbiased is:

If A and B are independent events, then which of the following statements are TRUE? (A) $P(A \cap B) = P(A).P(B)$ (B) $P(A \cap B) = P(A) - P(B)$ (C) $P(A \cup B) = P(A) + P(B) - P(A).P(B)$ (D) $P(A \cap B) = P(A). P(B|A)$ Choose the correct answer from the options given below:

Match List-I with List-II Let A and B be any two events | List-I | List-II | | --- | --- | | (A) $P(A')$ | (I) $\frac{P(A \cap B)}{P(A)}; P(A) \neq 0$ | | (B) $P(\phi)$ | (II) $\frac{P(A \cap B)}{P(B)}; P(B) \neq 0$ | | (C) $P(A\vert B)$ | (III) $1 - P(A)$ | | (D) $P(B\vert A)$ | (IV) 0 | Choose the correct answer from the options given below:

If X is a normal variate with mean 16 and standard deviation 4, then the value of standard normal variate Z corresponding to X = 17 is:

In 5 trials of binomial distribution, the probability of 3 successes is 4 times the probability of 2 successes. The probability of success in each trial is:

The standard deviation of the number of tails in three tosses of a coin is:

Three bad eggs are mixed with 7 good ones. If two eggs are drawn one by one without replacement, then the probability distribution of the number (X) of bad eggs drawn is: | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 1/4 | 1/2 | 1/4 | | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 15/61 | 20/61 | 26/61 | | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 7/15 | 7/15 | 1/15 | | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 1/8 | 1/4 | 5/8 |

The random variable $x$ has a probability distribution $p(x)$ of the form $P(x=r)=\begin{cases} rk, & \text{if } r \le 2, \\ (r-1)k, & \text{if } 2<r\le 4, \\ 0, & \text{otherwise,} \end{cases}$ where $r \in \mathbb{N}\cup\{0\}$ and $k\in\mathbb{R}$, where $\mathbb{N}$ is the set of natural numbers. Then: (A) $k=\frac{1}{9}$ (B) $P(2\le x\le 3)=\frac{1}{2}$ (C) $P(x=4)=\frac{1}{3}$ (D) $P(x>1)=\frac{7}{8}$ Choose the correct answer from the options given below.

In class XII, suppose 5% of boys and 0.25% of girls are physically fit for a game. A fit student is selected at random from this class having same number of boys and girls. If the probability that the selected student is a girl is $\frac{m}{n}$, gcd(m, n) = 1, then m + n is equal to

A random variable X has the following probability distribution | X | 0 | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|---| | P(X) | 0.1 | k | 0.2 | 2k | 0.3 | k | Then P (X < 3) is

A bag contains 4 red and 6 green balls. A ball is drawn at random. Its colour is noted and is returned to the bag. One additional ball of the colour drawn is put in the bag. Again a ball is then drawn from the bag. The probability of this ball to be of green colour is

A box contains 2 black and 4 red balls and another box contains 4 black and 3 red balls. If a ball drawn at random from one of the two boxes, then the probability of getting a black ball is

If the random variable X is normally distributed with mean 16 and standard deviation 4, then the standard normal variable Z corresponding to X = 17 is

A box contains 10 balls, each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, then the probability that none is marked with the digit 0 is:

In a game, a man wins ₹ 8 for getting a number greater than 3 and loses ₹ 3 otherwise, when a fair die is thrown. The man decided to throw a die 4 times but to quit as and when he gets a number greater than 3. If X denotes the amount which the man wins or loses, then which of the following are correct? (A) All the possible values of X are 8, 5, 2 and -1. (B) The probability distribution of X is: | X | 8 | 5 | 2 | -1 | -12 | |---|---|---|---|---|---| | P(X) | 1/2 | 1/4 | 1/8 | 1/16 | 1/16 | (C) The mean value of X is 75/16. (D) The variance of X is 6615/256. Choose the correct answer from the options given below:

If a random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | K | K/2 | K/4 | K/8 | then, Match List-I with List-II | List-I | List-II | |---|---| | (A) The value of K is | (I) 2/15 | | (B) P(0 < X < 2) is | (II) 1/15 | | (C) P(1 < X < 3) is | (III) 8/15 | | (D) P(X > 2) is | (IV) 4/15 | Choose the correct answer from the options given below:

If A speaks truth in 75% cases and B speaks truth in 80% cases, then the probability that they contradict each other in a statement, is:

If A and B are two events such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$, $P(B|A) = \frac{1}{4}$, then $P(A|B)$ is:

Two events A and B will be independent, then

A player participates in 3 matches against three teams T₁, T₂ and T₃.The probability of winning a match against teams T₁, T₂ and T₃ are 0.2, 0.3 and 0.9 respectively. If 'wins' can be regarded as independent events, then the probability that he (A) wins all the 3 matches is 0.054 (B) wins no match is 0.054 (C) wins exactly two matches is 0.348 (D) wins exactly one match is 0.542 Choose the correct answer from the options given below:

If X is a random variable and a, b are real numbers, then which of the following statements are correct? (A) $E[aX+b] = a E(X) + b$ (B) $Var (aX + b) = a^2 Var (X) + b$ (C) $Var (aX + b) = a Var (X)$ (D) $Var (X) = E(X^2) - [E(X)]^2$ Choose the correct answer from the options given below:

A die is tossed 6 times and getting "1 or 5" is considered a success. The probability of getting at least one success in six tosses is:

If the probability that an individual suffers a bad reaction from an injection of a given serum is 0.001. The probability that out of 2000 individuals, more than two individuals suffer from bad reaction is: [Given that $e^{-2} \approx 0.13534$]

If a random variable $X$ has the following probability distribution: | X | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X) | k | 2k | 3k | k² | 6k² | , then Match List-I with List-II | List-I | List-II | |---|---| | (A) k | (I) 3/7 | | (B) $P(X < 2)$ | (II) 6/49 | | (C) $P(X > 3)$ | (III) 1/7 | | (D) $P(2 \leq X \leq 3)$ | (IV) 22/49 | Choose the correct answer from the options given below:

Let x denote the number of doublets in three throws of a pair of dice with the following probability distribution. | x | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(x) | $\frac{25}{72}k$ | $\frac{15}{72}k$ | $\frac{3}{72}k$ | $\frac{1}{360}k$ | If value of k is equal to $\frac{m}{n} \cdot \gcd(m,n) = 1$, then $m + n$ is equal to

Match List-I with List-II if $P(A) = \frac{3}{7}$, $P(B) = \frac{4}{7}$ and $P(A \cup B) = \frac{5}{7}$ | List-I | List-II | | :--- | :--- | | (A) $P(A \cap B)$ | (I) $\dfrac{2}{3}$ | | (B) $P(A \mid B)$ | (II) $\dfrac{5}{7}$ | | (C) $P(B \mid A)$ | (III) $\dfrac{1}{2}$ | | (D) $P(A' \cup B')$ | (IV) $\dfrac{2}{7}$ | Choose the correct answer from the options given below:

Match List-I with List-II An urn contains 4 white and 3 red balls. In a random draw of three balls, the probability of | List-I | List-II | |---|---| | (A) No red ball is | (I) $\frac{12}{35}$ | | (B) Only 1 red ball is | (II) $\frac{1}{35}$ | | (C) Exactly 2 red balls is | (III) $\frac{4}{35}$ | | (D) no white ball is | (IV) $\frac{18}{35}$ | Choose the correct answer from the options given below:

If a computer code is correctly programmed, it gives 90% acceptable results. But if it is not correctly programmed, it gives only 40% acceptable results. From previous experience, it is observed that only 80% of codes are correctly programmed. If after a certain programming, the code gives 2 acceptable results, then the approximate probability that the code is correctly programmed is

If $2P(A) = P(B) = \frac{5}{13}$ and $P(A|B) = \frac{2}{5}$, then $P(A \cap B)$ is

Under which of the following conditions the Poisson distribution is the limiting case of, binomial distribution: (A) The number of trials is indefinitely large. (B) The probability of success for each trial is indefinitely small. (C) The product of the number of trials and the probability of success for each trial is finite. (D) The probability of success for each trial is indefinitely large. Choose the correct answer from the options given below:

If $X$ is a normal variate with mean 12 and standard deviation 4, then $P[X \ge 20]$ is: [Given that: $P[0 \le Z \le 2] = 0.4772]$

In a game, a man wins Rs. 5 for getting a number greater than 4 and loses Rs. 1 otherwise, when a fair dice is thrown. The man decided to throw a die thrice but to quit as and when he gets a number greater than 4. The expected value of the amount(in Rs.) he wins or loses is:

The probability of a man hitting a target is 1/2. How many times must he fire so that the probability of hitting the target at least once is more than 90%?

Let the random variable X represent the positive difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. Then probability $P(X \leq 3)$ is equal to

An urn I contains 3 white and 4 blue balls, while urn II contains 5 white and 6 blue balls. One ball is drawn at random from one of the urns and it is found to be white. The probability that it was drawn from urn II is

If a person A speaks the truth in 80% cases and the person B speaks the truth in 75% cases, then the probability that they contradict each other in a statement is

A problem in Mathematics is given to two students X and Y whose chances of solving it are $\frac{1}{3}$ and $\frac{1}{4}$ respectively. The probability that only X solves the problem, is:

If we take 8 identical slips of paper and write the number 0 on one of them, the number 1 on three of the slips, the number 2 on three of the slips and the number 3 on one of the slips. These slips are folded, put in a box and roughly mixed. One slip is drawn at random from the box. If X is the random variable denoting the number written on the drawn slip, the variance of X is:

The normal distribution curve is symmetrical about [$\mu$ = mean, $\sigma$= standard deviation]

In a game, a person is paid Rs. 2 if he gets all heads or all tails when three coins are tossed, and he will pay Rs. 2 if either one or two heads show. What can he expect to win on an average per game?

A lot of 50 watches is known to have 10 defective watches. If 8 watches are selected one by one with a replacement at random, then the probability that there will be at least one defective watch is:

The probability distribution of a random variable X is given by | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | $1 - 7a^2$ | $\frac{1}{2}a + \frac{1}{4}$ | $a^2$ | If $a > 0$, then $P(0 < x \leq 2)$ is equal to

A black and a red die are rolled simultaneously. The probability of obtaining a sum greater than 9, given that the black resulted in a 5 is

If A and B are any two events such that P(B) = P(A and B), then which of the following is correct

If A is any event associated with sample space and If $E_1, E_2, E_3$ are mutually exclusive and exhaustive events. Then which of the following are true? (A) $P(A) = P(E_1)P(E_2|A) + P(E_2)P(E_2|A) + P(E_3)P(E_3|A)$ (B) $P(A) = P(A|E_1)P(E_1) + P(A|E_2)P(E_2) + P(A|E_3)P(E_3)$ (C) $P(E_i|A) = \frac{P(A|E_i)P(E_i)}{\sum_{i=1}^3 P(A|E_i)P(E_i)}$, $i = 1,2,3$ (D) $P(A|E_i) = \frac{P(E_i|A)P(E_i)}{\sum_{i=1}^3 P(E_i|A)P(E_i)}$, $i = 1,2,3$ Choose the correct answer from the options given below:

Match List-I with List-II Let A & B are two events such that P(A)=0.8, P(B)=0.5, P(B|A)=0.4 | List-I | List-II | | :--- | :--- | | (A) $P(A \cap B)$ | (I) 0.2 | | (B) $P(A \mid B)$ | (II) 0.32 | | (C) $P(A \cup B)$ | (III) 0.64 | | (D) $P(A')$ | (IV) 0.98 | Choose the correct answer from the options given below:

What is the mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on 1 face?

How many minimum number of times must a man toss a fair coin so that the probability of having at least one head is more than 90%?

Let $F(z)$ be the cumulative density function of the standard normal variate $z$, then which of the following are correct? (A) $F(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^z e^{-\frac{z^2}{2}} dz$, $-\infty < z < \infty$ (B) $F(-z) = 1 - F(z)$ (C) $F(0) = 0$ (D) $F(\infty) = 1$ Choose the correct answer from the options given below:

The probability distribution of the random variable X is given by | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | 0.2 | k | 2k | 2k | The variance of the random variable X is

A random variable X has the following probability distribution | X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |---|---|---|---|---|---|---|---|---|---| | P(X) | a | 3a | 5a | 7a | 9a | 11a | 13a | 15a | 17a | Then the values of 'a' and P(0 < X < 5) respectively are

If A and B are independent events such that $P(A|B) = \frac{1}{3}$ and $P(B) = \frac{1}{2}$, then the value of $P(A \cap B)$ is equal to

The probability of not getting 53 Sundays in a leap year is

If A and B are two events such that $P(A|B) = P(B|A)$, and $A \cap B \neq \phi$ then

A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is four. The probability that it is actually four is

In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is

Which of the following statements are NOT correct about Standard Normal Distribution? (A) The probability curve of the Standard Normal Distribution is a bell-shaped curve. (B) The Standard Normal variate (Z) score describes the position of each data point in terms of its distance from the mean, when measured in standard deviation units. (C) The Z-score is negative if the data point lies above the mean, and positive if it lies below the mean. (D) There is a 95.45 % probability of randomly selecting a score between $\mu - \sigma$ and $\mu + \sigma$, when $\sigma$ is standard deviation and $\mu$ is mean. Choose the correct answer from the options given below:

Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Match List-I with List-II | List-I | List-II | |---|---| | **X** | **Probability, P(X)** | | (A) 4 | (I) $\frac{1}{6}$ | | (B) 5 | (II) $\frac{5}{36}$ | | (C) 6 | (III) $\frac{1}{12}$ | | (D) 7 | (IV) $\frac{1}{9}$ | Choose the correct answer from the options given below:

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | 0.1 | 0.2 | 0.3 | 0.4 | The variance of the X will be:

A coin is tossed twice and outcomes are recorded. If the random variable X represents the number of heads in the experiment, then the expectation of X will be:

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---|---|---| | P(X) | 0 | k | 2k | 2k | 3k | k² | 2k² | 7k² + k | The value of $P(4 < x < 7)$ is equal to

Let E and F are events associated with an experiment. If $P(E) = 0.4$, $P(F) = 0.8$ and $P(F|E) = 0.6$, then $P(E|F)$ is

Which of the following statements are correct? (A) If E and F are independent events then $P(E \cap F) = P(E) \cdot P(F)$ (B) If E and F are mutually exclusive events, then $P(E \cup F) = P(E) + P(F) - P(E) \cdot P(F)$ (C) The conditional probability of an event E, given the occurrence of the event F is given by $\frac{P(E \cap F)}{P(F)}, P(F) \neq 0$ (D) If E and F be the events associated with the sample space S of an experiment, then $P(\overline{E}|F) = 2 - P(E|F)$ Choose the correct answer from the options given below:

One person speaks truth in 60% of the cases and another person in 80% of the cases. They are likely to agree in stating the same fact in

A die is thrown three times. If the first throw is a five, the probability of getting 14 as the sum is

The probability distribution of a ramdom variable $X$ is: $P(X=x)=\begin{cases}kx^2,& x=1,2,3\\2kx,&x=4,5,6\\0,&\text{otherwise}\end{cases}$ Where $K$ is a constant Match List-I with List-II | List-I | List-II | |---|---| | (A) $k$ | (I) 7/22 | | (B) $P(X \geq 4)$ | (II) 1/44 | | (C) $P(X < 4)$ | (III) 95/22 | | (D) $E[X]$ | (IV) 15/22 | Choose the correct answer from the options given below:

If three balls are drawn one by one without replacement from a bag containing 5 white and 4 red balls, then the probability distribution of the number of white balls drawn is

If $y$ is normal distribution random variable with mean $\mu = 10$ and standard deviation $\sigma = 2$. $z$ is standard normal variable and $F(Z)$ is cumulative distribution function, then which of the following are true? [Given that $F(1.5) = 0.9332$, $F(3) = 0.9986$, $F(2.25) = 0.9878$ and $F(1) = 0.8413$] (A) $P(X < 13) = 0.9332$ (B) $P(X > 16) = 0.9986$ (C) $P(12 < X < 14.5) = 0.1465$ (D) $P(X > 8) = 0.8413$ Choose the correct answer from the options given below:

If the probability of two successes is 9 times the probability of 3 successes in 3 trials of a binomial distribution, then the probability of success in each trial is:

A die is rolled thrice. What is the probability of getting a number greater than $4$ in the first and the second throw of dice and a number less than $4$ in the third throw ?

Two dice are thrown simultaneously. If X denotes the number of fours, then the expectation of X will be :

If the random variable $ X $ has the following distribution: | $X$ | 0 | 1 | 2 | otherwise | | --- | --- | --- | --- | --- | | $P(X)$ | $ k $ | $ 2k $ | $ 3k $ | $ 0 $ | Match List-I with List-II: | List-I | List-II | | --- | --- | | (A) $ k $ | (I) $ \frac{5}{6} $ | | (B) $ P(X < 2) $ | (II) $ \frac{4}{3} $ | | (C) $ E(X) $ | (III) $ \frac{1}{2} $ | | (D) $ P(1 \leq X \leq 2) $ | (IV) $ \frac{1}{6} $ |

Let $X$ denote the number of hours you play during a randomly selected day. The probability that $X$ can take values $x$ has the following form, where $c$ is some constant. $\mathrm{P}(\mathrm{X}=\mathrm{x})=\left\{\begin{array}{lll} 0.1, & \text { if } \mathrm{x}=0 \\ \mathrm{cx}, & \text { if } \mathrm{x}=1 \text { or } \mathrm{x}=2 \\ \mathrm{c}(5-\mathrm{x}), & \text { if } \mathrm{x}=3 \text { or } \mathrm{x}=4 \\ 0, & \text { otherwise } \end{array}\right.$ Match List-I with List-II : | List-I | List-II | | --- | --- | | (A) $ c $ | (I) 0.75 | | (B) $ P(X \leq 2) $ | (II) 0.3 | | (C) $ P(X = 2) $ | (III) 0.55 | | (D) $ P(X \geq 2) $ | (IV) 0.15 |

There are two bags. Bag-1 contains $4$ white and $6$ black balls and Bag-2 contains $5$ white and $5$ black balls. A die is rolled, if it shows a number divisible by 3, a ball is drawn from Bag-1, else a ball is drawn from Bag-2. If the ball drawn is not black in colour, the probability that it was not drawn from Bag-2 is :

The probability of not getting $53$ Tuesdays in a leap year is :

The probability of a shooter hitting a target is $\frac34$. How many minimum number of times must he fire so that the probability of hitting the target at least once is more than $90 \%$ ?

Three defective bulbs are mixed with $8$ good ones. If three bulbs are drawn one by one with replacement, the probabilities of getting exactly $1$ defective, more than $2$ defective, no defective and more than $1$ defective respectively are:

There are $6$ cards numbered $1$ to $6$, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two cards drawn. Then $\mathrm{P}(\mathrm{X}>3)$ is :

A coin is tossed K times. If the probability of getting $3$ heads is equal to the probability of getting $7$ heads, then the probability of getting $8$ tails is :

If a fair coin is tossed 10 times, then the probability of obtaining at least one head is :

Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Then variance of the number of kings is

A doctor is to visit a patient. It is known that the probabilities that he will come by train, bus, scooter or by other means of transport are respectively $\frac{3}{10}, \frac{1}{5}, \frac{1}{10}$ and $\frac{2}{5}$. The probabilities that he will be late are $\frac{1}{4}, \frac{1}{3}$ and $\frac{1}{12}$, if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he arrives late. The probability that he comes by bus is:

If a fair coin is tossed 10 times the probability of atleast 6 heads is: