Algebra

Probability

Medium

applied

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: $e^{-7.5} \approx 0.000553$ ]

0.00412

0.00312

0.00415

0.00315

✅ Correct Option: 3

The defective rate is 3%, so the probability of a defective item is $p = 0.03$ .

The packet contains $n = 250$ items.

Given that $e^{-7.5} \approx 0.000553$ , this suggests using Poisson approximation since $n$ is large, $p$ is small, and the exponential term is provided.

For Poisson approximation, calculate $\lambda$ :

$\lambda = n \times p$

$\lambda = 250 \times 0.03$

$\lambda = 7.5$

The Poisson formula for exactly $k$ occurrences is:

$P(X = k) = \dfrac{\lambda^k \times e^{-\lambda}}{k!}$

For exactly 1 defective item, substitute $k = 1$ and $\lambda = 7.5$ :

$P(X = 1) = \dfrac{7.5^1 \times e^{-7.5}}{1!}$

$P(X = 1) = \dfrac{7.5 \times 0.000553}{1}$

$P(X = 1) = 0.0041475$

$P(X = 1) \approx 0.00415$

Therefore, the probability that a packet of 250 items contains exactly one defective item is $0.00415$ .

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Match List-I with List-II

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X	Probability, P(X)
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(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:

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≈0.000553e^{-7.5} \approx 0.000553e−7.5≈0.000553]

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

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: $e^{-7.5} \approx 0.000553$ ]