/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Car Fatalities The recent rate o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 15

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 in the next billion miles is approximately 0.9999877, indicating it is almost certain there will be at least one fatality.

Step by step solution

01

Understand the Given Data

First, note that the rate of car fatalities is 33,561 fatalities per 2969 billion miles.
02

Calculate the Fatality Rate

Find the rate of fatalities per billion miles. This can be calculated as follows: \( \text{Rate} = \frac{33561}{2969} \approx 11.3 \text{ fatalities per billion miles} \)
03

Interpreting Poisson Distribution

The problem involves a rare event (car fatalities), so it can be modeled with a Poisson distribution where \( \text{Rate} = \text{Lambda} = 11.3 \).
04

Define the Poisson Probability Formula

The formula for the Poisson probability is \( P(X=k) = \frac{\text{Lambda}^k e^{-\text{Lambda}}}{k!} \).
05

Calculate the Probability for At Least One Fatality

To find the probability of at least one fatality, we calculate the complement of the probability of zero fatalities: \( P(X eq 0) = 1 - P(X=0) \). So, \( P(X=0) = \frac{11.3^0 e^{-11.3}}{0!} = e^{-11.3} = 1.23 \times 10^{-5} \).
06

Final Calculation

Subtract the probability of zero fatalities from 1: \( P(X eq 0) = 1 - 1.23 \times 10^{-5} \). The probability of at least one fatality in the next billion miles is approximately 0.9999877.
07

Interpret the Result

The probability that there will be at least one fatality in the next billion miles traveled is extremely high, which indicates that it is almost certain that at least one fatality will occur.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Poisson Distribution
The Poisson Distribution is ideal for modeling rare or random events within a fixed interval of time or space, such as car fatalities over a certain number of miles. This distribution is defined by its average rate, often called 'Lambda' (位).

The formula to determine the probability of observing exactly k events in a Poisson distribution is:

\( P(X=k) = \frac{\text{Lambda}^k e^{-\text{Lambda}}}{k!} \)

Key points to remember:
  • \(\lambda\) represents the average number of events within the interval.
  • e is the base of natural logarithms, approximately equal to 2.71828.
  • k! is the factorial of k.
The Poisson distribution is especially useful in situations where events are independent and occur at a constant average rate.
Fatality Rate Calculation
To calculate the fatality rate, you need to divide the total number of fatalities by the total number of miles traveled (or other relevant unit).

In the given solution, the fatality rate is calculated as:

\( \text{Rate} = \frac{33561}{2969} \approx 11.3 \text{ fatalities per billion miles} \)

This means, on average, there are 11.3 fatalities for every billion miles traveled. Understanding this rate is critical for further calculations and in applying the Poisson distribution to predict the likelihood of future events.
Probability of Events
The probability of events helps us understand the likelihood of certain events occurring within a given context. Using the Poisson distribution, we can calculate the probability of observing a specific number of fatalities.

For example, if we want to find the probability of having 0 fatalities, we use:

\( P(X=0) = \frac{11.3^0 e^{-11.3}}{0!} = e^{-11.3} = 1.23 \times 10^{-5} \)

This small probability indicates that having zero fatalities is highly unlikely. By understanding this, we can make more informed decisions about travel safety and risk management.
Complement Rule in Probability
The Complement Rule in Probability states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring.

In this exercise, we are interested in finding the probability of at least one fatality. We calculate this using the complement rule:

\( P(X eq 0) = 1 - P(X=0) \)

Given our previous calculation:
\( P(X=0) = 1.23 \times 10^{-5} \)
we get:

\( P(X eq 0) = 1 - 1.23 \times 10^{-5} \approx 0.9999877 \)

This very high probability shows that it is nearly certain there will be at least one fatality in the next billion miles traveled. The complement rule is a valuable tool in determining the likelihood of alternative outcomes in probability.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

News Source Based on data from a Harris Interactive survey, \(40 \%\) of adults say that they prefer to get their news online. Four adults are randomly selected. a. Use the multiplication rule to find the probability that the first three prefer to get their news online and the fourth prefers a different source. That is, find \(P(\mathrm{OOOD})\), where \(\mathrm{O}\) denotes a preference for online news and D denotes a preference for a news source different from online. b. Beginning with \(\mathrm{OOOD}\), make a complete list of the different possible arrangements of those four letters, then find the probability for each entry in the list. c. Based on the preceding results, what is the probability of getting exactly three adults who prefer to get their news online and one adult who prefers a different news source.

Significant For 100 births, \(P\) (exactly 56 girls \()=0.0390\) and \(P(56\) or more girls) \(=0.136\). Is 56 girls in 100 births a significantly high number of girls? Which probability is relevant to answering that question?

Assume that when adults with smartphones are randomly selected, \(54 \%\) use them in meetings or classes (based on data from an LG Smartphone survey). If 12 adult smartphone users are randomly selected, find the probability that fewer than 3 of them use their smartphones in meetings or classes.

a. Grades \((\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{F})\) earned in statistics classes b. Heights of students in statistics classes c. Numbers of students in statistics classes d. Eye colors of statistics students e. Numbers of times statistics students must toss a coin before getting heads

Assume that when adults with smartphones are randomly selected, \(54 \%\) use them in meetings or classes (based on data from an LG Smartphone survey). If 20 adult smartphone users are randomly selected, find the probability that exactly 15 of them use their smartphones in meetings or classes.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.