Applied Mathematics

Probability Distribution

Medium

applied

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

$\frac{(50)^5 \times e^{-5}}{50!}$

$\frac{5^{50}\times e^{-5}}{50!}$

$\frac{5^{50}\times e^{-50}}{50!}$

$\frac{5^5\times e^{-50}}{50!}$

✅ Correct Option: 2

With $n=50$ and $p=0.1$ , use the Poisson approximation with $\lambda = np = 5$ .

$P(X=50) = \frac{e^{-\lambda}\lambda^{50}}{50!} = \frac{5^{50}e^{-5}}{50!}$ .

2026: 29 May Shift 2

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 $e^{-2} =0.1353$ ]

2026: 22 May Shift 2

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.

(D) There are 35 students whose height is greater then 72 inches.

Choose the correct answer from the options given below:

2026: 30 May Shift 1

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:

2026: 29 May Shift 2

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:

2023: 11 Aug Shift 1

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

The value of k is

2026: 30 May Shift 2

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:

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

Solution

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