Question

The probability of a component being defective is 0.01. There are 100 such components in a machine. Then the probability of two or more defective components in the machine is _______

• A. \(1 - e^{-1}\)
• B. \(2e^{-1}\)
• C. \(1 - 2e^{-1}\)
• D. \(e^{-1}\)

Hint:

Use Poisson approximation when \(n\) is large, \(p\) is small: \( \lambda = np \), and subtract lower probabilities.

Answer 1

The Correct Option is C

Given: \[ n = 100, p = 0.01 \Rightarrow np = 1 \Rightarrow \lambda = 1 \] This is a Poisson distribution problem with mean \(\lambda = 1\)
We need \( P(X \geq 2) = 1 - P(0) - P(1) \)
\[ P(0) = \frac{e^{-1} (1)^0}{0!} = e^{-1}, P(1) = \frac{e^{-1} (1)^1}{1!} = e^{-1} \] \[ P(X \geq 2) = 1 - e^{-1} - e^{-1} = 1 - 2e^{-1} \] Final Answer: (3) \(1 - 2e^{-1}\)