Question: A grinding mill breaks down on average $\lambda = 5$ times a month. What is $P(X = 0)$ = probability it will _not_ break down in a month?
1. **State the problem:** We want to find the probability that the grinding mill does not break down at all in a month, given the average breakdown rate $\lambda = 5$ times per month.
2. **Identify the distribution:** The number of breakdowns $X$ follows a Poisson distribution with parameter $\lambda = 5$.
3. **Poisson probability formula:**
$$
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
$$
where $k$ is the number of occurrences (breakdowns).
4. **Calculate $P(X=0)$:**
$$
P(X=0) = \frac{5^0 e^{-5}}{0!} = e^{-5}
$$
5. **Evaluate the expression:**
$$
P(X=0) = e^{-5} \approx 0.0067
$$
6. **Interpretation:** The probability that the mill will not break down in a month is approximately $0.0067$, or about $0.67\%$.