Problem 34 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The following table gives the number of people killed in rollover crashes in various types of vehicles in 2002 : Find the empirical probability distribution associated with these data. If a fatality due to a rollover crash in 2002 is picked at random, what is the probability that the victim was in a. \(\mathrm{A}\) car? b. An SUV? c. A pickup or an SUV?

Step-by-Step Solution

Answer

a. The probability that the victim was in a car is approximately \(P(Car) = 0.5418\). b. The probability that the victim was in an SUV is approximately \(P(SUV) = 0.2783\). c. The probability that the victim was in a pickup or an SUV is approximately \(P(Pickup\ or\ SUV) = 0.5907\).

Step 1: Find the total number of deaths

To find the empirical probability distribution, we first need to find the total number of deaths. We will use this as the denominator in our probabilities. Total deaths = Number of deaths in Cars + Number of deaths in Pickups + Number of deaths in SUVs + Number of deaths in Vans Total deaths = 4768 + 2742 + 2448 + 698

Step 2: Calculate the empirical probabilities

Now, we calculate the probability for each type of vehicle. P(Car) = Deaths in Cars / Total deaths P(Pickup) = Deaths in Pickups / Total deaths P(SUV) = Deaths in SUVs / Total deaths P(Van) = Deaths in Vans / Total deaths

Step 3: Find the probability that the victim was in a car

Use the total deaths and the number of deaths in cars to find P(Car). P(Car) = 4768 / (4768 + 2742 + 2448 + 698)

Step 4: Find the probability that the victim was in an SUV

Use the total deaths and the number of deaths in SUVs to find P(SUV). P(SUV) = 2448 / (4768 + 2742 + 2448 + 698)

Step 5: Find the probability that the victim was in a pickup or an SUV

Add the probabilities P(Pickup) and P(SUV) to find the probability that the victim was in a pickup or an SUV. P(Pickup or SUV) = P(Pickup) + P(SUV) P(Pickup or SUV) = (2742 / (4768 + 2742 + 2448 + 698)) + (2448 / (4768 + 2742 + 2448 + 698)) Now, calculate the probabilities and we get the empirical probability distribution for a fatality due to a rollover crash in 2002 is picked at random: a. Probability the victim was in a car: P(Car) = 0.5418 (approximately) b. Probability the victim was in an SUV: P(SUV) = 0.2783 (approximately) c. Probability the victim was in a pickup or an SUV: P(Pickup or SUV) = 0.5907 (approximately)

Key Concepts

Probability Calculation

Probability calculation involves determining how likely an event is to occur. In empirical probability calculations, this involves gathering real-world data and analyzing the frequencies of different outcomes. To calculate the probability of an event, you divide the number of successful outcomes by the total number of possible outcomes.
In the context of vehicle rollover fatalities, we're determining how likely it is that a fatality occurred in a certain type of vehicle. For example:

• The probability, P(Car), is calculated by dividing the number of deaths in cars by the total number of fatalities. If there are 4768 car fatalities out of 10656 total fatalities, then P(Car) = 4768 / 10656.
• This results in a probability of about 0.5418, meaning there's a 54.18% chance the victim was in a car.

These simple calculations are powerful tools in understanding risks and outcomes associated with different events.

Empirical Probability

Empirical probability is based on actual data or observed events rather than theoretical probability, which is based on what is expected to happen in an ideal scenario. It involves calculating probabilities from past data to make predictions about future events.
For instance, by analyzing vehicle safety statistics from 2002, one can determine the probability of fatalities by vehicle type during that year. This data-driven approach finds its application in many fields like insurance, healthcare, and transportation.
The core steps include:

• Collecting reliable and accurate data relevant to the problem.
• Counting the frequencies of observed events, such as the number of fatalities in different vehicle types.
• Calculating the probabilities as the ratio of the event frequency to the total number of occurrences.

This method gives insight into the likelihood of future events, helping stakeholders make informed decisions.

Data Analysis

Data analysis is crucial in transforming raw data into meaningful insights. When it comes to calculating empirical probabilities, data analysis helps in correctly organizing and interpreting the data, ensuring accurate probability estimations.
In our exercise, analyzing the data involved summing up total fatalities, categorizing them by vehicle type, and then calculating the specific probabilities such as for cars, SUVs, or pickups.
Steps in data analysis typically include:

• Verification: Ensuring the integrity and reliability of data before computation.
• Computation: Calculating totals and averages to understand the data's central tendencies.
• Interpretation: Drawing conclusions from the data by looking at calculated probabilities and understanding what they imply about real-world risks.

An effective data analysis strategy allows us to handle large data sets, handle anomalies attentively, and derive accurate conclusions that are essential for making evidence-based decisions.

Vehicle Safety Statistics

Understanding vehicle safety statistics can significantly impact decisions related to policy making, insurance, and public safety initiatives. Such statistics shed light on critical areas requiring intervention to increase safety.
The data from 2002 highlights the number of fatalities associated with different vehicle types in rollover crashes. Such statistics can be used to determine safety ratings, influence manufacturer designs, and guide administrative guidelines.
Key aspects include:

• Identifying risk factors: Understanding which vehicle types are most prone to fatal rollovers.
• Informing policy: Using statistics to support or propose legislation that enhances road safety, like stricter car designs or mandatory safety features.
• Educating the Public: Providing insights that guide consumers towards safer driving practices and choices.

By thoroughly analyzing these statistics, stakeholders can work collaboratively to mitigate risks and enhance overall vehicle safety.