The probability distribution of a random variable X is given as P(X=x)=caseskx^3 & for x=1,2,3\\ 3kx & for x=4,5\\ 0 & otherwisecases, where k is a constant. The value of P(X≥4) is Option 1: 5/21 Option 2: 16/21 Option 3: 3/7 Option 4: 4/7

Answer: 3/7. Solution: Total probability: k(1+8+27)+3k(4+5)=36k+27k=63k=1, so k=1/63. Then P(X≥4)=12k+15k=27k=27/63=37.

Applied Mathematics

Probability Distribution

Medium

applied

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

$\dfrac{5}{21}$

$\dfrac{16}{21}$

$\dfrac{3}{7}$

$\dfrac{4}{7}$

✅ Correct Option: 3

Total probability: $k(1+8+27)+3k(4+5)=36k+27k=63k=1$ , so $k=\dfrac{1}{63}$ . Then $P(X\ge4)=12k+15k=27k=\dfrac{27}{63}=\dfrac37$ .

2026: 18 May Shift 2

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:

2026: 18 May Shift 2

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

2023: 7 June Shift 1

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

2026: 14 May Shift 2

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:

2026: 15 May Shift 1

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

2026: 21 May Shift 2

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

Choose the correct answer from the options given below: