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Chapter 4: Problem 43

Small cars get better gas mileage, but they are not as safe as bigger cars. Small cars accounted for \(18 \%\) of the vehicles on the road, but accidents involving small cars led to 11,898 fatalities during a recent year (Reader's Digest, May 2000 ). Assume the probability a small car is involved in an accident is .18. The probability of an accident involving a small car leading to a fatality is .128 and the probability of an accident not involving a small car leading to a fatality is .05. Suppose you learn of an accident involving a fatality. What is the probability a small car was involved? Assume that the likelihood of getting into an accident is independent of car size.

Short Answer

Expert verified
The probability is approximately 35.97% that a small car was involved in the accident given a fatality occurred.

Step by step solution

01

Identify the Known Probabilities

We are given the following probabilities:- The probability that a small car is involved in any accident, \(P(S) = 0.18\).- The probability that an accident involving a small car leads to a fatality, \(P(F|S) = 0.128\).- The probability that an accident not involving a small car leads to a fatality, \(P(F|N) = 0.05\), where \(N\) represents not a small car.
02

Use Complement Rule for Non-Small Cars

The probability that an accident involves a non-small car is the complement of the probability for a small car:\[ P(N) = 1 - P(S) = 1 - 0.18 = 0.82 \].
03

Calculate Total Probability of a Fatality

To find the total probability of a fatality, use the law of total probability:\[ P(F) = P(F|S)P(S) + P(F|N)P(N) \].Substitute the known values:\[ P(F) = 0.128 \times 0.18 + 0.05 \times 0.82 \]. Compute this to get the total probability of a fatality.
04

Calculate the Total Probability analytically

Calculate the total probability:\[ P(F) = 0.128 \times 0.18 + 0.05 \times 0.82 = 0.02304 + 0.041 = 0.06404 \].
05

Apply Bayes' Theorem

Use Bayes' theorem to find the probability that a small car was involved given a fatality:\[ P(S|F) = \frac{P(F|S) \times P(S)}{P(F)} \].Substitute the known values:\[ P(S|F) = \frac{0.128 \times 0.18}{0.06404} \].
06

Compute Bayes' Theorem

Calculate:\[ P(S|F) = \frac{0.02304}{0.06404} \approx 0.3597 \]. Thus, there is approximately a 35.97% probability that a small car was involved in the accident given a fatality occurred.

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Key Concepts

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

Understanding Probability
Probability is a fundamental mathematical concept that helps us understand the likelihood of various events. It's expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 signifies a certain event.
In our example, we're dealing with accidents involving small cars. The probability that a small car is involved in any accident is given as 0.18. This means that out of all car accidents, 18% involve small cars.
Probabilities are often given in percentage form to enhance comprehension. For instance, if you know that the probability a small car will be in an accident is 0.18, you can also express this as 18%.
  • Probabilities help in forecasting and risk assessment.
  • They can guide decision-making by evaluating potential outcomes.
  • Understanding probabilities is key in various fields, from finance to medicine.
Exploring Conditional Probability
Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. This concept is vital for understanding scenarios where outcomes are dependent on certain conditions being met.
In the problem at hand, we use conditional probability to determine the likelihood of a fatality given that a small car is involved in an accident. This is represented as \(P(F|S)\), which means the probability of a fatality given a small car was involved in the accident.
  • In this scenario, \(P(F|S) = 0.128\), meaning there's a 12.8% chance that an accident involving a small car results in a fatality.
  • Similarly, \(P(F|N) = 0.05\) indicates a 5% probability of a fatality in accidents not involving small cars.
Conditional probabilities allow us to refine our predictions by taking specific circumstances into account, enhancing the precision of our conclusions.
Applying the Law of Total Probability
The law of total probability provides a way to calculate the overall likelihood of an event by considering all possible ways that event can occur. It incorporates conditional probabilities for each scenario and their respective probabilities.
In our example, to find the total probability of a fatality (\(P(F)\)), we consider both the probability of fatalities involving small cars \(P(F|S)\) and those not involving small cars \(P(F|N)\).
  • Calculate \(P(F)\) using: \[ P(F) = P(F|S)P(S) + P(F|N)P(N) \]
  • Substitute the known values: \[ P(F) = 0.128 \times 0.18 + 0.05 \times 0.82 \]
  • The result, \(P(F) = 0.06404\), represents the overall probability of a fatality occurring in any given car accident.
This law helps in evaluating complex scenarios by breaking them down into simpler sub-events, allowing us to consider all contributing factors in calculating probabilities.

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Most popular questions from this chapter

The U.S. Department of Transportation reported that during November, \(83.4 \%\) of Southwest Airlines flights, \(75.1 \%\) of US Airways flights, and \(70.1 \%\) of JetBlue flights arrived on time (USA Today, January 4, 2007). Assume that this on-time performance is applicable for flights arriving at concourse A of the Rochester International Airport, and that \(40 \%\) of the arrivals at concourse A are Southwest Airlines flights, \(35 \%\) are US Airways flights, and \(25 \%\) are JetBlue flights. a. Develop a joint probability table with three rows (airlines) and two columns (on-time arrivals vs. late arrivals). b. An announcement has just been made that Flight 1424 will be arriving at gate 20 in concourse A. What is the most likely airline for this arrival? c. What is the probability that Flight 1424 will arrive on time? d. Suppose that an announcement is made saying that Flight 1424 will be arriving late. What is the most likely airline for this arrival? What is the least likely airline?

Simple random sampling uses a sample of size \(n\) from a population of size \(N\) to obtain data that can be used to make inferences about the characteristics of a population. Suppose that, from a population of 50 bank accounts, we want to take a random sample of four accounts in order to learn about the population. How many different random samples of four accounts are possible?

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A study of 31,000 hospital admissions in New York State found that \(4 \%\) of the admissions led to treatment-caused injuries. One-seventh of these treatment-caused injuries resulted in death, and one-fourth were caused by negligence. Malpractice claims were filed in one out of 7.5 cases involving negligence, and payments were made in one out of every two claims. a. What is the probability a person admitted to the hospital will suffer a treatment-caused injury due to negligence? b. What is the probability a person admitted to the hospital will die from a treatmentcaused injury? c. In the case of a negligent treatment-caused injury, what is the probability a malpractice claim will be paid?

The U.S. Census Bureau provides data on the number of young adults, ages \(18-24,\) who are living in their parents' home. ' Let \(M=\) the event a male young adult is living in his parents' home \(F=\) the event a female young adult is living in her parents' home If we randomly select a male young adult and a female young adult, the Census Bureau data enable us to conclude \(P(M)=.56\) and \(P(F)=.42\) (The World Almanac, 2006 ). The probability that both are living in their parents' home is .24 a. What is the probability at least one of the two young adults selected is living in his or her parents' home? b. What is the probability both young adults selected are living on their own (neither is living in their parents' home)?
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