Theoretical DistributionsMCQMTP Dec 2023 Series IQuestion 3523 of 230
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If 5%\displaystyle 5\% of the electric bulbs manufactured by a company are defective, use Poisson distribution to find the probability that in a sample of $100bulbs,\displaystyle bulbs,5$ bulbs will be defective. [Given : $e^{-5} = 0.007$ ]

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option a — $0.1823$

All Options:

  • A$0.1823$
  • B$0.1723$
  • C$0.1623$
  • D$0.1923$

Detailed Solution & Explanation

Let n=100\displaystyle n = 100 be the sample size, and let p=5%=0.05\displaystyle p = 5\% = 0.05 be the probability of a bulb being defective. The mean parameter m\displaystyle m for the Poisson distribution is: m=np=100×0.05=5m = np = 100 ×\displaystyle \times 0.05 = 5 We need to find the probability of exactly 5\displaystyle 5 defective bulbs, i.e., P(X=5)\displaystyle P(X = 5). The probability mass function of the Poisson distribution is: P(X=k)=e−mmkk!P(X = k) = e−mmkk!\displaystyle \frac{e^{-m} m^k}{k!} Substituting m=5\displaystyle m = 5 and k=5\displaystyle k = 5 into the formula: P(X=5)=e−5555!=e−5×3125120=e−5×62524P(X = 5) = \displaystyle \frac{e^{-5} 5^5}{5!} = \frac{e^{-5} ×\displaystyle \times 3125}{120} = e^{-5} ×\displaystyle \times 62524\displaystyle \frac{625}{24} Using the given approximation e−5=0.007\displaystyle e^{-5} = 0.007: P(X=5)≈0.007×26.04167≈0.1823P(X = 5) ≈\displaystyle \approx 0.007 ×\displaystyle \times 26.04167 ≈\displaystyle \approx 0.1823 This corresponds to Option A. Note: The source option key indicates Option D, but the correct mathematical calculation yields the probability as 0.1823, which is Option A. Hence, **Option A** is the correct answer.

About This Chapter: Theoretical Distributions

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Binomial, Poisson, Normal Distribution

This chapter covers Binomial, Poisson, Normal Distribution and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

Key Concepts to Understand

Probability

A numerical measure of the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). Formula: P(A) = Number of favorable outcomes / Total number of equally likely outcomes.

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