CUET Mathematics Applied Mathematics > Probability Distribution PYQs

There are 10% defective pens. In a sample of randomly selected 10 pens, the probability that atmost one pen is defective, is

Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) Mean of a normal variate X is 12 and standard deviation is 4, then the Z-score of data point 20 is | (I) 1 | | (B) If X is Poisson variate such that P(X=1) = 2P(X=2), then the mean of X is | (II) 8 | | (C) In a binomial distribution, if mean is 5 and variance is 4, then the number of trials is | (III) 2 | | (D) In a binomial distribution, if n = 20 and q = 0.6, then its mean is | (IV) 25 | Choose the correct answer from the options given below:

In a printing press of a publication house, there is a small chance $\frac{1}{1000}$ for any typing error in 20 pages, then the number of pages containing no error in printing two volumes of an Encyclopedia containing total 10000 pages are: [Given: $e^{-0.02}=0.9802$]

A box containing 10 wall clocks has three defective pieces in it. If random sample of two clocks is taken from the box, then the probability distribution of defective clocks is:

On a multiple choice examination with four possible answers (out of which only one is correct) for each of five questions, what is the probability that a candidate would get four or more correct answers just by guessing?

The number of telephone calls made daily in a certain community between 8 P.M. and 9 P.M. has a mean of 352 and standard deviation of 31. What percentage of time will there be more than 400 telephone calls made in this community between 8 P.M. to 9 P.M.? [Given that: $P(0\leq Z\leq1.55)=0.4394$, where $Z$ is the standard normal variate]

A box contains 8 red and 2 white balls. If three balls are drawn, one by one, at random without replacement, then the variance of the number of white balls drawn is:

If 0.1% of the bolts manufactured by a machine are found to be defective. Find the probability that, in a sample of 2000 bolts chosen at random, exactly 3 will be defective using the Poisson distribution. [Given $e^{-2} =0.1353$]

In a binomial distribution, the sum of its mean and variance is 1.8. If the event was conducted 5 times, then the probability of 3 successes is

The probability distribution of a discrete random variable is given below : | X | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| | P(X) | 1/5 | m | 2/5 | 2m | 3m | Which of the following are correct? A. $P(2)=\frac{1}{15}$ B. $P(4)=\frac{2}{15}$ C. $P(X<2)=\frac{1}{5}$ D. $P(X\geq4)=\frac{1}{5}$ Choose the correct answer from the options given below:

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

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X) | 0.1 | 0.15 | 0.30 | 0.30 | 0.15 | Which of the following are correct? (A) $P(X \le 1)=0.25$ (B) $P(X>3)=0.15$ (C) $P(X=2)=0.30$ (D) $P(X \le 2)=0.55$ Choose the correct answer from the options given below:

If X is a Poisson variate such that $3P(X=2)=2P(X=1)$, then $P(X=0)$ is equal to: (Given $e^{-4/3}=0.264$)

Let X denote the number of scores in a test. If X is normally distributed with mean 100 and standard deviation 15, then the probability that X does not exceed 130 is, (Given $P(0 \le Z \le 2)=0.4772$, where $Z$ be the standard normal variate)

A traffic police officer records the number of motorcycle riders that use a particular lane on a highway. He records that an average of 2.5 motorcycle riders use the lane every hour. Given that the number of motorcycles that use the particular lane follows a Poisson distribution. The probability that less than 2 motorcycle riders will use the lane within an hour will be:- (Use $e^{2.5} = 12.2$)

In a random experiment, a collection of trials is called Bernoulli trials then which of the following statements are TRUE? A. The number of trials is infinite. B. Each trial has exactly two outcomes defined as success and failure. C. The trials are dependent by nature. D. The probability of success remains the same in each trial. Choose the correct answer from the options given below:

If $X$ is a random variable such that $X$ can take values 0,1, 2 or 3. Then the expectation of $X$ for the following data is : | $X$ | 0 | 1 | 2 | 3 | |---|---|---|---|---| | $P(X)$ | $k$ | $k^2$ | $1-5k^2$ | $k^2$ | (where $k>0$)

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

The mean and variance of a random variable $X$ having a binomial distribution are 1 and 2/3 respectively. Then, Match the LIST-I with LIST-II | LIST-I | LIST-II | |---|---| | A. $P(X=0)$ | I. $1/27$ | | B. $P(X=3)$ | II. $2/9$ | | C. $P(X=2)$ | III. $4/9$ | | D. $P(X=1)$ | IV. $8/27$ | Choose the correct answer from the options given below:

If X is a random variable which can take values 0,1,2,3 such that $E(X^2) = 2E(X)$. If $P(X=0) = P(X=1) = m\ \&\ P(X=2) = 2m$, then which of the following are TRUE? A. $P(X=1) = \dfrac{1}{13}$ B. $P(X=3) = \dfrac{1}{13}$ C. $E(X) = \dfrac{18}{13}$ D. $P(X=2) = \dfrac{6}{13}$ Choose the correct answer from the options given below:

If the heights of 300 students are normally distributed with mean 68 inches and standard deviation 3 inches. If Z is the standard normal variate such that $P(0\le Z\le 1.33)=0.4082, P(Z\ge 1.33)=0.0918$ and $P(0\le Z\le 1)=0.3413$, then which of the following statements are correct? (A) There are 27 students whose height is greater than 72 inches. (B) The number of students having heights less than or equal to 64 inches is 27. (C) 210 students have their heights between 65 and 71 inches. (D) There are 35 students whose height is greater then 72 inches. Choose the correct answer from the options given below:

The probability that a man aged 35 years will die before reaching the age of 40 years may be taken as 0.018. Out of a group of 100 men, now aged 35 years, what is the approximate probability that one man will die within next five years? (Given: $e^{-1.8}=0.1653$)

A random variable X follows Poisson distribution. The variance of X is 2. The value of $P(X=3)$ is

If a random variable X follows binomial distribution with mean $\dfrac{4}{3}$ and variance $\dfrac{8}{9}$. Which of the following are correct? A. Probability of success is $\dfrac{1}{3}$. B. Probability of failure is $\dfrac{2}{3}$. C. Number of trials is 4. D. $P(X \ge 1) = \dfrac{35}{81}$ Choose the correct answer from the options given below:

In a single throw of a die, if X denotes the number on its upper face, then the mean of X is,

The random variable X has a probability distribution $P(X)$ of the following form, where $k$ is some real number. $P(X=x) = \begin{cases}k, & \text{if } x=0\\\dfrac{k}{2}, & \text{if } x=1 \text{ or } 2\\0, & \text{otherwise}\end{cases}$ On the basis of the above information, match List-I with List-II | LIST-I | | LIST-II | |---|---|---| | A. Value of $k$ | | I. $\dfrac{1}{4}$ | | B. $P(X<2)$ | | II. 0 | | C. $P(X>1)$ | | III. $\dfrac{1}{2}$ | | D. $P(X>2)$ | | IV. $\dfrac{3}{4}$ | Choose the correct answer from the options given below:

A car hire firm has two cars, which is hire out day by day. The number of demands for cars on each day is distributed as a Poisson distribution with mean 1.5. Then which of the following is/are TRUE? (use $e^{-1.5} = 0.2231$) A. The probability of days when no car is required is 0.2231 B. The probability of days when some demand is refused is 0.1913 C. The probability of days when no car is required is 0.1913 D. The probability of days when some demand is refused is 0.2231 Choose the correct answer from the options given below:

If $X$ is a discrete random variable which assumes values $x_1, x_2, x_3, ..., x_n$ with the respective probabilities $p_1, p_2, ....., p_n$, then Match the LIST-I with LIST-II | | LIST-I | | LIST-II | |---|---|---|---| | A. | $\displaystyle\sum_{i=1}^{n} p_i =$ | I. | $E(X^2)$ | | B. | $\displaystyle\sum_{i=1}^{n} p_i x_i =$ | II. | $Var(X)$ | | C. | $\displaystyle\sum_{i=1}^{n} p_i x_i^2 =$ | III. | $1$ | | D. | $E(X^2) - \{E(X)\}^2 =$ | IV. | $E(X)$ | Choose the correct answer from the options given below:

The mean and variance of a Binomial distribution are 4 and 4/3 respectively. The number of trials is

A pair of dice is thrown 5 times. If getting a total of 7 is considered a success, then Match the LIST-I with LIST-II | LIST-I | LIST-II | |---|---| | A. Probability of no success | I. $26\left(\dfrac{1}{6}\right)^5$ | | B. Probability of 5 successes | II. $2\left(\dfrac{5}{6}\right)^5$ | | C. Probability of at least 4 successes | III. $\left(\dfrac{1}{6}\right)^5$ | | D. Probability of at most one success | IV. $\left(\dfrac{5}{6}\right)^5$ | Choose the correct answer from the options given below:

For 6 trials of an experiment, let X be a binomial variate which satisfies the relation $9P(X=4)=P(X=2)$, then the probability of success is

If $X$ is a Poisson variate such that $P(X=3)=2P(X=4)$, then $P(X=2)$ equals: [given $e^{-2}=0.14$]

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

Which of the following statements are correct? (A) The mean of a binomial distribution is always greater than the variance. (B) The mean of a Poisson distribution is always greater than the variance. (C) If $f(x)$ is the probability density function of a random variable $X$, then $P(a \leq X \leq b) = \int_a^b f(x)dx$. (D) If $X$ is a random variable and $a,b$ are real numbers, then $Var(aX+b)=a^2Var(X)$. Choose the correct answer from the options given below:

Let $X$ be a discrete random variable whose probability distribution is defined as follows; $P(X=x)=\begin{cases}k(x+1), & x=1,2,3,4\\ 2kx, & x=5,6,7\\ 0, & \text{otherwise}\end{cases}$ where $k$ is constant, then $P(X=5)$ is equal to

A coin is tossed until a head appears, or the tail appears 4 times in succession, then the probability distribution of the number of tosses is

The variance of the number of heads in 16 tosses of a coin is

Let the random variable $X$ follow a Poisson distribution. If $P(X=3)=\dfrac{2}{5}P(X=2)$, then $P(X=1)$ is equal to

The probability distribution of a random variable X is given as: $P(X=x) = \begin{cases} kx^2 & for & x=1,2,3 \\ 2kx & for & x=4,5,6 \\ 0 & otherwise \end{cases}$ Where $k$ is an arbitrary constant, then $E(X)$ is equal to:

If X has a Poisson distribution such that $2P(X=1)=P(X=2)$, then which of the following are correct? A. The mean of X is 4 B. The mean of X is 2 C. $P(X=2)=\frac{8}{e^4}$ D. $P(X=3)=\frac{32}{e^4}$ Choose the correct answer from the options given below:

In a random experiment, a collection of trials is called Bernoulli trials if A. The number of trials is infinite. B. The trials are independent of each other. C. Each trial has exactly two outcomes, defined as success and failure. D. The probability of success will vary in each trial. Choose the correct answer from the options given below:

Match the LIST-I with LIST-II | | LIST-I | | LIST-II | |---|---|---|---| | A. | In a binomial distribution, if $n = 16$, $p = 0.75$, then the mean is | I. | $4$ | | B. | In a binomial distribution, if $n = 25$, $p = \dfrac{1}{5}$, then variance is | II. | $32$ | | C. | In a binomial distribution, the mean is 8 and the variance is 6, then the number of trials is | III. | $18$ | | D. | If the mean and variance of a binomial distribution are 6 and 4 respectively, then the number of trials is | IV. | $12$ | Choose the correct answer from the options given below:

If $X$ is a Poisson variate such that $4P(X=1)=3P(X=2)$, then $P(X=0)=$

The probability distribution of a random variable $X$ is given as $P(X=x)=\begin{cases}kx^3 & \text{for } x=1,2,3\\ 3kx & \text{for } x=4,5\\ 0 & \text{otherwise}\end{cases}$, where $k$ is a constant. The value of $P(X\ge4)$ is

If a coin is tossed 8 times and getting a head is considered as success, then which of the following statements are correct ? (A) The probability of getting 4 heads is $\dfrac{35}{128}$ (B) The mean of the number of successes is 4. (C) The variance of the number of successes is 2. (D) The probability of getting 6 heads is $\dfrac{9}{64}$. Choose the **correct** answer from the options given below:

Ajay is a vendor who informs that there are 2% rotten apples in the fruit bag. Find the probability that a sample of 10 apples will include not more than one rotten apple.

If the random variable X follows the Poisson distribution such that P[X = k] = P[X = k+1], then the mean value of X is:

| $ X $ | -2 | -1 | 0 | 1 | 2 | | --- | --- | --- | --- | --- | --- | | $ P(X) $ | $ 0.2 $ | $ 0.1 $ | $ 0.3 $ | $ 0.2 $ | $0.2$ | The variance of $X$ will be :

The variance of the number of heads in two tosses of a coin is :

10 works hard drawn successively with replacement from a lot containing 10% defective bulb. The probability that there is at least one defective bulb is :

The probability distribution of a discrete random variable X is given by : | X | 30 | 10 | -10 | |---|---|---|---| | P(X) | 1/5 | 3/10 | 1/2 | then E(X) is equals to

2 voices dice are thrown together. For the first die $P(6) = \frac{1}{2}$, other scores are equally likely. While for the second die $P(1) = \frac{2}{5}$ and other scores are equally likely than the Mean for the probability distribution of the number of one score will be

In a binomial distribution the probability of getting success is $\frac{1}{4}$ And standard deviation is 3 then its mean is ?

$P(X = x) = \begin{cases} 2k & \text{if } x = 0 \\ kx & \text{if } x = 1 \\ k(x - 1) & \text{if } x = 2 \text{ or } 3 \\ 0 & \text{otherwise} \end{cases}$ The value of k is

A random variable X has a probability distribution P(X) of the following form, where k is some unknown constant: P(X = 0) = k P(X = 1) = 2k P(X = 2) = 3k P(X = other values) = 0 Then, find the value of 1/k.

Match List - I with list- II | List-I | List - II | |---|---| | A. the probability distribution is applied for discrete random variable | normal distribution | | B. A normal distribution is symmetric about | standard deviation | | C. this probability distribution is applied for continuous random variable | mean | | D. the shape of normal curve depend upon | Poisson distribution | Choose the correct option below :

Match list I with list II. 4 defective pens are mixed with 10 normal pens. 3 pens are drawn one by one with replacement , then the probability distribution of the number of defective pens is : | List-I | List - II | |---|---| | A. P(X=0) | 8/343 | | B. P(X=1) | 60/343 | | C. P(X=2) | 125/343 | | D. P(X=3) | 150/343 | Choose the correct option below :

If the mean of a binomial distribution is 24 and its standard deviation is 4 , then the probability of getting success is :

The random variable X has a probability distribution P(X) of the following form where k is a scalar and $P(X = x) = \begin{cases} k, & \text{if } x = 0 \\ 2k, & \text{if } x = 1 \\ 3k, & \text{if } x = 2 \\ 0, & \text{otherwise} \end{cases}$ then value of P(X < 2) = _______.

A book consisting of 2000 pages has 540 misprints distributed randomly throughout the book. The average number of misprints in one page of the book is:

The variance of the Binomial Distribution $B\left(5, \frac{1}{4}\right)$ is:

A shopkeeper wants to check the average number of cars sold per call. Past record of sales is shown below: | Sale of cars (Units) | 0 | 1 | 2 | 3 | |---|---|---|---|---| | Probabilities | $\frac{1}{6}$ | $\frac{1}{2}$ | $\frac{3}{10}$ | $\frac{1}{30}$ | The expected number of cars sold is :

If the mean of a binomial distribution is 12 and its standard deviation is 2, then the number of trials is :

If the probability distribution of X is : | X | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---| | P(X) | 1/15 | 2/15 | 3/15 | 4/15 | 5/15 | Then variance is equal to :

If a random variable X follows binomial distribution with mean 5 and variance $\frac{5}{2}$, then $P(X \leq 9)$ is :

Which of the following statements are correct ? (A) $\text{var}(aX + b) = a^2 \text{var}(X)$ (B) $\text{var}(X) = E(X^2) - \{E(X)\}^2$ (C) $E(aX + b) = aE(X) + b$ (D) $E(X) = \sum_{i=1}^{n} p_i x_i^2$ Choose the correct answer from the options given below :

In binomial distribution with $n = 10$ and $P = \frac{1}{3}$, the probability of the event that unequal number of failures and successes occur is :

The probability distribution of a random variable X is given below : | X | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X) | 0.1 | 0.25 | 0.3 | 0.2 | 0.15 | Then, $\text{Var}\left(\frac{X}{2}\right)$ is :

If X has a Poisson distribution such that $P(X = 1) = P(X = 2)$ then $P(X = 3)$ is :

A discrete random variable X has the following probability distribution : | X: | 0 | 1 | 2 | 3 | 4 | 5 | |----|----|----|----|----|----|----| | P(X): | b | 3b | 5b | 3b | 4b | 6b | The value of b is :

A discrete random variable X takes the values 0, 1, 2, 3, 4 and its mean is 1.6. If $P(X=1) = 0.4$, $P(X=4) = P(X=2)$ and $P(X=3) = 2P(X=2)$, then $P(X=0)$ is :

A telephone exchange receives on an average 5 calls per minute. The probability of receiving 3 or less calls per minute is :

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) In a binomial distribution, if $n = 10$, $q = 0.25$, then its mean is | (I) 12 | | (B) If the mean of a binomial distribution is 6 and its variance is 3, then p is | (II) 7.5 | | (C) In a binomial distribution, the probability of getting a success is $\frac{1}{4}$ and the standard distribution is 3, then its mean is | (III) 16 | | (D) If the mean and variance of a binomial distribution are 4 and 3 respectively, then the number of trials is | (IV) $\frac{1}{2}$ | Choose the correct answer from the options given below :

If the sum and product of the mean and variance of a binomial distribution are 18 and 72 respectively, then the probability of obtaining atmost one success is

Between 3 p.m. and 5 p.m. the average number of phone calls per minute coming into the helpline desk of a bank is 5. The probability that during one particular minute there will be only one phone call is :

Match List I with List II | LIST I | LIST II | |---|---| | A. The variance of a Poisson distribution with mean $\lambda$ is | I. $\sqrt{\lambda}$ | | B. The standard deviation of a Poisson distribution with mean $\lambda$ is | II. 4 | | C. In a Poisson distribution, if mean is 4, then the standard deviation is | III. $\lambda$ | | D. In a Poisson distribution, if mean is 4, then the variance is | IV. 2 | Choose the correct answer from the options given below:

If the probability distribution of a discrete random variable $X$ is given as | X | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X) | 0.5 | 2k | 3k | 3k | 2k | Then the value of $k$ is:

If the probability distribution of a random variable X is given as | $x_i$ | 0 | 1 | 2 | 3 | |---|---|---|---|---| | $p_i$ | $2k^2$ | $k^2$ | $3k^2$ | $k$ | Then the mean of X is