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The problem involves finding Poisson probabilities related to homicide data over several city-years. From the image, the key parameters are as follows:

- There are 280 city-years in total.
- There were 73 homicides in total.

### Step 1: Find the mean number of homicides per city-year.
The mean (λ) is calculated as the total number of homicides divided by the number of city-years:
\[
\lambda = \frac{73}{280} = 0.2607
\]

### Step 2: Poisson probability formula
The Poisson probability formula is:
\[
P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!}
\]
Where:
- \( k \) is the number of homicides.
- \( \lambda = 0.2607 \)
- \( e \) is the Euler's number (approximately 2.71828).

### Step 3: Calculation for each value of \( k \)
We already have probabilities for \( k = 0 \), \( k = 1 \), and \( k = 2 \), so we can calculate for \( k = 3 \).

#### a. \( P(0) \)
\[
P(0) = \frac{e^{-0.2607} \cdot 0.2607^0}{0!} = e^{-0.2607} \approx 0.7705 \quad \text{(given)}
\]

#### b. \( P(1) \)
\[
P(1) = \frac{e^{-0.2607} \cdot 0.2607^1}{1!} \approx 0.2009 \quad \text{(given)}
\]

#### c. \( P(2) \)
\[
P(2) = \frac{e^{-0.2607} \cdot 0.2607^2}{2!} \approx 0.0262 \quad \text{(given)}
\]

#### d. \( P(3) \)
\[
P(3) = \frac{e^{-0.2607} \cdot 0.2607^3}{3!}
\]
Let's calculate this:
\[
P(3) = \frac{e^{-0.2607} \cdot (0.2607)^3}{6} = \frac{e^{-0.2607} \cdot 0.01773}{6} \approx \frac{0.7705 \cdot 0.01773}{6} \approx \frac{0.01367}{6} \approx 0.0023
\]

Thus, \( P(3) \approx 0.0023 \).

### Summary of probabilities:
- \( P(0) = 0.7705 \)
- \( P(1) = 0.2009 \)
- \( P(2) = 0.0262 \)
- \( P(3) = 0.0023 \)

Would you like more details or have any further questions?

---

Here are five related questions to consider:
1. How do Poisson distributions apply to real-world crime data analysis?
2. What assumptions are made when using a Poisson distribution for modeling homicides?
3. How does changing the mean \( \lambda \) affect the shape of the Poisson distribution?
4. What would the probabilities look like for values of \( k \geq 4 \)?
5. How could the accuracy of these probability models be verified with further data?

**Tip:** When working with small means like \( \lambda = 0.2607 \), the probabilities drop off quickly for higher values of \( k \), which is why \( P(3) \) and higher are very small.

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: 0, 1, 2, or 3, using the Poisson probabilities.

This page explores how to use the Poisson distribution to calculate probabilities for homicide occurrences in 14 cities over a 20-year period. Learn how to find the mean number of homicides per city-year, and how to apply the Poisson formula to determine probabilities for 0, 1, 2, or 3 homicides in a given year. The solution uses key concepts from probability theory and statistics.

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