Answer: 0.0506

## Explanation This is a Poisson distribution probability calculation problem with a rate conversion from monthly to yearly. **Key Steps:** 1. **Monthly Rate:** The firm receives clients at a rate of λ = 4 clients per month. 2. **Yearly Rate:** Since there are 12 months in a year, the yearly rate is: $$\lambda_{year} = 4 \times 12 = 48$$ 3. **Poisson Probability Formula:** For a Poisson distribution with rate λ, the probability of exactly x events is: $$P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}$$ 4. **Calculation:** We need to find P(X = 44) with λ = 48: $$P(X = 44) = \frac{e^{-48} \times 48^{44}}{44!}$$ 5. **Result:** This calculation yields approximately 0.0506. **Why other options are incorrect:** - **A (0.025):** This might be an approximation error or using incorrect parameters. - **C (0.24):** This is too high and doesn't match the Poisson distribution calculation. - **D (0.00363):** This could result from using the monthly rate (λ = 4) instead of converting to yearly rate. **Conceptual Understanding:** - The Poisson distribution is appropriate for modeling the number of events occurring in a fixed interval of time when events occur independently at a constant average rate. - When converting time periods, the rate parameter scales linearly with time (monthly rate × 12 = yearly rate). - The probability mass function accounts for both the expected number of events (λ) and the specific count we're interested in (x).

Author: Nikitesh Somanthe