Question:
The lifetime (in years) of bulbs is distributed as an \( \text{Exp}(1) \) random variable. Using Poisson approximation to the binomial distribution, the probability (rounded off to 2 decimal places) that out of the fifty randomly chosen bulbs at most one fails within one month equals
The lifetime (in years) of bulbs is distributed as an \( \text{Exp}(1) \) random variable. Using Poisson approximation to the binomial distribution, the probability (rounded off to 2 decimal places) that out of the fifty randomly chosen bulbs at most one fails within one month equals
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When using Poisson approximation, the mean number of events is \( \mu = np \), and the Poisson distribution can be used to approximate the probability of a specific number of events occurring.
Updated On: Dec 12, 2025
- 0.05
- 0.07
- 0.09
- 0.11
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The Correct Option is C
Solution and Explanation
Step 1: Understand the exponential distribution.
The lifetime of each bulb follows an exponential distribution with rate parameter \( \lambda = 1 \). The probability that a bulb fails within one month is \( P(\text{failure in 1 month}) = 1 - e^{-1/12} \). Step 2: Use Poisson approximation.
The number of failures in 50 bulbs follows a Poisson distribution with parameter \( \mu = 50 \cdot P(\text{failure in 1 month}) \). Using the Poisson distribution, we approximate the probability of at most one failure: \[ P(X \leq 1) = P(X = 0) + P(X = 1) \] Substituting the appropriate values, we find \( P(X \leq 1) \approx 0.07 \). Step 3: Conclusion.
Thus, the correct answer is (B) 0.07.
The lifetime of each bulb follows an exponential distribution with rate parameter \( \lambda = 1 \). The probability that a bulb fails within one month is \( P(\text{failure in 1 month}) = 1 - e^{-1/12} \). Step 2: Use Poisson approximation.
The number of failures in 50 bulbs follows a Poisson distribution with parameter \( \mu = 50 \cdot P(\text{failure in 1 month}) \). Using the Poisson distribution, we approximate the probability of at most one failure: \[ P(X \leq 1) = P(X = 0) + P(X = 1) \] Substituting the appropriate values, we find \( P(X \leq 1) \approx 0.07 \). Step 3: Conclusion.
Thus, the correct answer is (B) 0.07.
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