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To find the probability \( P(0) \) using the Poisson distribution, we need to determine the mean number of homicides per city-year, \( \lambda \), which is the average rate of occurrences.

### Step 1: Calculate the mean \( \lambda \)
Given:
- Total homicides: 73
- Total city-years: 280

The mean \( \lambda \) is calculated as:
\[
\lambda = \frac{\text{Total homicides}}{\text{Total city-years}} = \frac{73}{280} \approx 0.2607
\]

### Step 2: Apply the Poisson formula
The Poisson probability formula is:
\[
P(x) = \frac{e^{-\lambda} \lambda^x}{x!}
\]
where:
- \( e \approx 2.71828 \)
- \( x = 0 \) (since we are calculating \( P(0) \))
- \( \lambda \approx 0.2607 \)

\[
P(0) = \frac{e^{-0.2607} \cdot 0.2607^0}{0!} = e^{-0.2607}
\]

Now, I'll calculate \( e^{-0.2607} \).The probability \( P(0) \) is approximately \( 0.7705 \) when rounded to four decimal places.

Would you like to see further details or have any questions about this calculation? Here are a few related questions for deeper understanding:

1. How would the probability change if the mean number of homicides (\( \lambda \)) increased?
2. Can you calculate the probabilities \( P(1) \), \( P(2) \), etc., using the same method?
3. How does the Poisson distribution handle situations with rare events, like this one?
4. Why is the Poisson distribution suitable for modeling this scenario?
5. What would the expected number of city-years with zero homicides be, given the actual results?

**Tip:** The Poisson distribution is ideal for modeling the probability of a given number of events happening in a fixed interval when the events occur independently and at a constant average rate.

Data from 14 cities were combined for a 20-year period, and the total 280 city-years included a total of 73 homicides. After finding the mean number of homicides per city-year, find the probability that a randomly selected city-year has the following numbers of homicides, then compare the actual results to those expected by using the Poisson probabilities. P(0) is requested.

In this problem, data from 14 cities over a 20-year period are analyzed using the Poisson distribution. With 73 total homicides and 280 city-years, we calculate the mean number of homicides per city-year and find the Poisson probability of a city-year having zero homicides. Learn how to apply the Poisson formula to find P(0) and understand expected outcomes.

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