In a certain U.S. state, the rate of motor vehicle theft is 605 thefts per 100,000 vehicles. A large parking structure in the state capital has issued 491 parking permits. Note: The vehicles may or may not be stolen from the parking structure. Hint: Is it appropriate to use the Poisson approximation to the binomial distribution? If so, use the Poisson approximation.
What is the probability that at least one of the vehicles with a permit will eventually be stolen? (Round your answer to four decimal places.)


To solve this problem, we can use the **Poisson approximation** to the **binomial distribution**. This approximation is appropriate when the number of trials \( n \) is large, and the probability of success \( p \) is small. Here's the approach:

### Step 1: Define the variables
- The rate of motor vehicle theft is \( 605 \) thefts per \( 100,000 \) vehicles, which gives us the probability that a given vehicle will be stolen:
  \[
  p = \frac{605}{100,000} = 0.00605
  \]
- The number of vehicles with parking permits is \( n = 491 \).

### Step 2: Conditions for using Poisson approximation
The Poisson approximation to the binomial distribution is valid when:
- \( n \) is large (which it is here, since \( n = 491 \)).
- \( p \) is small (which it is here, since \( p = 0.00605 \)).

### Step 3: Use the Poisson approximation
The Poisson approximation says that the number of thefts \( X \) can be approximated by a Poisson distribution with parameter \( \lambda = n \cdot p \), where \( \lambda \) is the expected number of successes.

\[
\lambda = 491 \times 0.00605 = 2.97055
\]

Thus, \( X \sim \text{Poisson}(\lambda = 2.97055) \).

### Step 4: Find the probability of at least one theft
We need to find the probability that at least one vehicle will be stolen, i.e., \( P(X \geq 1) \). This can be computed as:

\[
P(X \geq 1) = 1 - P(X = 0)
\]

For a Poisson distribution, the probability of \( X = 0 \) is given by:

\[
P(X = 0) = \frac{\lambda^0 e^{-\lambda}}{0!} = e^{-\lambda}
\]

Substitute the value of \( \lambda \):

\[
P(X = 0) = e^{-2.97055} \approx 0.0513
\]

So, the probability of at least one vehicle being stolen is:

\[
P(X \geq 1) = 1 - 0.0513 = 0.9487
\]

### Final Answer:
The probability that at least one of the vehicles with a permit will be stolen is approximately **0.9487** (rounded to four decimal places).

---

Would you like more details on the Poisson approximation or binomial distribution? Here are 5 related questions to expand your understanding:

1. How does the Poisson distribution differ from the binomial distribution?
2. In what other real-world scenarios can we use the Poisson distribution?
3. How would the probability change if the number of parking permits increased?
4. What if the theft rate was 1,200 thefts per 100,000 vehicles instead?
5. How do we calculate the expected number of vehicles stolen?

**Tip:** The Poisson distribution is particularly useful for modeling rare events occurring over time or space, such as system failures or accidents.

In a certain U.S. state, the rate of motor vehicle theft is 605 thefts per 100,000 vehicles. A large parking structure in the state capital has issued 491 parking permits. Note: The vehicles may or may not be stolen from the parking structure. Hint: Is it appropriate to use the Poisson approximation to the binomial distribution? If so, use the Poisson approximation. What is the probability that at least one of the vehicles with a permit will eventually be stolen? (Round your answer to four decimal places.)

Learn how to calculate the probability that at least one of 491 permitted vehicles in a U.S. parking structure will be stolen, using the Poisson approximation to the binomial distribution. This problem involves a vehicle theft rate of 605 per 100,000 vehicles, probability theory, and the Poisson distribution. The solution is suitable for college-level students studying probability and statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation