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Chapter 5: Problem 16

Car Fatalities The recent rate of car fatalities was 33,561 fatalities for 2969 billion miles traveled (based on data from the National Highway Traffic Safety Administration). Find the probability that for the next billion miles traveled, there will be at least one fatality. What does the result indicate about the likelihood of at least one fatality?

Short Answer

Expert verified
The probability of at least one fatality per billion miles is approximately 99.99876%. This indicates an extremely high likelihood of at least one fatality.

Step by step solution

01

- Understand the problem

We need to find the probability that there will be at least one fatality per billion miles traveled, based on the given rate of car fatalities.
02

- Determine the rate of fatalities

The problem states there are 33,561 fatalities for 2969 billion miles traveled. Therefore, the rate of fatalities per billion miles is: \[ \text{Fatality rate} = \frac{33,561}{2969} \text{ fatalities per billion miles} \]
03

- Calculate the fatality rate

Calculate the rate: \[ \text{Fatality rate} = \frac{33,561}{2969} \text{ fatalities per billion miles} \ \text{Fatality rate} \ = 11.3 \text{ fatalities per billion miles} \]
04

- Use Poisson distribution

The Poisson distribution is suitable for this problem because it models the number of events (fatalities) that occur within a fixed interval (next billion miles). The Poisson probability of at least one fatality is: \[ P(X eq 0) = 1 - P(X = 0) \] where \( X \) is the number of fatalities.
05

- Calculate the probability of zero fatalities

Using the Poisson distribution formula: \[ P(X = 0) = \frac{\text{e}^{-\text{rate}} \times \text{rate}^0}{0!} \ P(X = 0) = \text{e}^{-11.3} \ P(X = 0) \ = 1.234 \times 10^{-5} \]
06

- Calculate the probability of at least one fatality

The probability of at least one fatality is: \[ P(X eq 0) = 1 - P(X = 0) \ P(X eq 0) = 1 - 1.234 \times 10^{-5} \ P(X eq 0) \ \text{approximately} = 0.9999876 = 99.99876\% \]
07

- Conclusion

This result indicates that there is an extremely high likelihood (almost 100%) that there will be at least one fatality in the next billion miles traveled.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Poisson Distribution
The Poisson distribution is a probability model that is often used for counting the number of events that happen in a fixed interval of time or space. It's particularly useful when these events are rare and independent of each other. In our car fatalities example, the 'event' is a car fatality, and the 'fixed interval' is one billion miles traveled. The formula for the Poisson distribution is:

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Most popular questions from this chapter

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