Recent crime reports indicate that 5.7 motor vehicle thefts occur each minute in the United States. Assume that the distribution of thefts per minute can be approximated by the Poisson probability distribution. Calculate the probability exactly five thefts occur in a minute.





To calculate the probability of exactly five motor vehicle thefts occurring in one minute using the Poisson probability distribution, we can use the formula:

\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
\]

where:
- \( \lambda \) is the average number of events (in this case, thefts) in a given time period,
- \( k \) is the number of events we want the probability for (here, 5 thefts),
- \( e \) is the base of the natural logarithm (approximately 2.71828).

Given:
- \( \lambda = 5.7 \)
- \( k = 5 \)

Now, substituting these values into the formula:

\[
P(X = 5) = \frac{5.7^5 e^{-5.7}}{5!}
\]

Calculating the components:

1. Calculate \( 5.7^5 \):
   \[
   5.7^5 \approx 18807.674
   \]

2. Calculate \( e^{-5.7} \):
   \[
   e^{-5.7} \approx 0.003351
   \]

3. Calculate \( 5! \) (factorial of 5):
   \[
   5! = 120
   \]

Now, plug these values back into the formula:

\[
P(X = 5) = \frac{18807.674 \times 0.003351}{120}
\]

Calculating this:

\[
P(X = 5) \approx \frac{63.110}{120} \approx 0.5259
\]

Therefore, the probability of exactly five thefts occurring in a minute is approximately **0.5259**, or **52.59%**.

### Summary:
The probability of exactly five motor vehicle thefts occurring in one minute is about **52.59%**.

Would you like more details on the Poisson distribution or have any other questions? Here are a few related questions you might find interesting:

1. How does the Poisson distribution differ from other probability distributions?
2. What is the significance of the parameter \( \lambda \) in the Poisson distribution?
3. How can we calculate the probability of fewer than five thefts occurring in a minute?
4. What real-world applications utilize the Poisson distribution?
5. Can the Poisson distribution be used for events with a different average rate? 

**Tip:** Always verify that the conditions for using the Poisson distribution are met, particularly the independence of events and a constant average rate.

Recent crime reports indicate that 5.7 motor vehicle thefts occur each minute in the United States. Assume that the distribution of thefts per minute can be approximated by the Poisson probability distribution. Calculate the probability exactly five thefts occur in a minute.

Explore how to calculate the probability of exactly five motor vehicle thefts occurring in one minute using the Poisson distribution. With an average theft rate of 5.7 per minute, learn about the key formulas and concepts involved. Ideal for students in Grades 11-12.

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