Question

If on average 4 customers visit a shop in an hour, then the probability that more than 2 customers visit the shop in a specific hour is:

• A. \( \frac{e^4 - 13}{e^4} \)
• B. \( \frac{8}{e^4} \)
• C. \( \frac{4}{e^4} \)
• D. \( \frac{e^4 - 21}{e^4} \)

Hint:

Use Poisson distribution for rare event probability calculations, particularly in time-based problems.

Answer 1

The Correct Option is A

Step 1: Define Poisson distribution.
The Poisson probability mass function (PMF) is: \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \] where \( \lambda = 4 \). Step 2: Compute probability for \( X \leq 2 \).
\[ P(X = 0) = \frac{e^{-4} 4^0}{0!} = e^{-4} \] \[ P(X = 1) = \frac{e^{-4} 4^1}{1!} = 4e^{-4} \] \[ P(X = 2) = \frac{e^{-4} 4^2}{2!} = 8e^{-4} \] \[ P(X \leq 2) = e^{-4} + 4e^{-4} + 8e^{-4} = 13e^{-4} \] Step 3: Compute probability for \( X>2 \).
\[ P(X>2) = 1 - P(X \leq 2) = 1 - 13e^{-4} \] Thus, the required probability is: \[ \mathbf{\frac{e^4 - 13}{e^4}} \]