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UPSC NDA I 2022 Mathematics Solved Paper

Question Numbers: 115-117

Consider the following data for the next three (03) items that follow :

The incidence of suffering from a disease among workers in an industry has a chance of

33\frac{1}{3}\%.

Question : 115 of 120

Marks: +1, -0

What is the probability that exactly 3 out of 6 workers suffer from a disease?

Solution:

Concept:

Use binomial distribution because each worker has independent probability

p = \frac{1}{3}

of suffering from the disease.

Explanation:

Given:

33\frac{1}{3}\% = \frac{1}{3}

probability for a worker to have the disease.

Let

p = \frac{1}{3}

and

q = 1 - p = \frac{2}{3}

Number of workers

n = 6

. We need exactly

x = 3

workers with the disease.

Binomial probability formula:

P(X = x) = \binom{n}{x} p^x q^{n-x}

Thus,

P(X = 3) = \binom{6}{3} \left(\frac{1}{3}\right)^3 \left(\frac{2}{3}\right)^{3}

Compute:

\binom{6}{3} = 20

20 \times \frac{1}{27} \times \frac{8}{27} = 20 \times \frac{8}{729} = \frac{160}{729}

Answer:

\frac{160}{729}

(Option D).

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