1. **Problem Statement:** We want to understand the Poisson distribution, which models the number of events occurring in a fixed interval of time or space when these events happen with a known constant mean rate and independently of the time since the last event. 2. **Formula:** The probability of observing exactly $k$ events is given by the Poisson probability mass function: $$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$$ where: - $\lambda$ is the average number of events in the interval, - $k$ is the number of events we want to find the probability for, - $e$ is Euler's number, approximately 2.71828. 3. **Important Rules:** - $k$ must be a non-negative integer ($k = 0, 1, 2, \ldots$). - The events are independent. - The average rate $\lambda$ is constant. 4. **Example Calculation:** Suppose the average number of emails received in an hour is $\lambda = 3$. What is the probability of receiving exactly 5 emails in an hour? Using the formula: $$P(X=5) = \frac{3^5 e^{-3}}{5!}$$ Calculate step-by-step: - $3^5 = 243$ - $5! = 120$ - $e^{-3} \approx 0.0498$ So, $$P(X=5) = \frac{243 \times 0.0498}{120} = \frac{12.1014}{120} \approx 0.1008$$ 5. **Interpretation:** There is about a 10.08% chance of receiving exactly 5 emails in an hour when the average is 3. This method can be used to find probabilities for any number of events $k$ given the average rate $\lambda$.