## Auto Insurance and Binomcdf Functions

### Topic

Auto Insurance and Binomcdf Functions

### Vocabulary

- Binomial Distributions
- Binomial Probability
- Binomcdf Function

### Student Exploration

#### Is your auto insurance expensive? How do auto insurance companies determine how much to charge someone for car insurance?

Insurance companies use complex equations to measure the risk of insuring each individual, whether it is for homeowner insurance, renter’s insurance, auto insurance, etc. Insurance provides a guarantee of compensation for a specific loss, damage, illness or death. For example, homeowner insurance can help cover the cost of your belongings if you were to have a fire, accident, flood, earthquake, or be robed; such that you can replace the items or make repairs so that your house is in the same condition as prior to the incident. Car insurance covers you in the case of a car accident, vandalism, or theft. Binomial probabilities allow us to calculate the measure the likelihood of one event occurring, and similarly the probability of it not occurring. And you can use the Cumulative Distribution Function (CDF) for Binomial Distributions, also called the “Binomcdf” function on your TI-84 calculator to find the probability of less than or equal to a number of successes out of a certain number of trials.

**Real World Example**: An auto insurance company provides insurance coverage for 150 people and on average a person makes a claim one every four months. What is the probability that no more than 5 people will make claims in in these four months?

To find the binomcdf function, divide each month into n intervals being one day. We know that there are 120 days in four months. Thus the probability that any one person will make a claim on any given day is \begin{align*}\frac{1}{120}\end{align*}, which is approximately equal to 0.0083333. We can now set up our function

\begin{align*}P(X < 5) = {_{150}}C_5 \left(\frac{1}{120}\right)^5 \left(\frac{119}{120}\right)^{145}\end{align*}

You also know that we can use our TI-84 calculator to determine this value. We enter it as: binomcdf \begin{align*}\left(150, \frac{1}{120}, 5 \right)\end{align*}. We find that binomcdf\begin{align*}\left(150, \frac{1}{120}, 5 \right) = 0.99827\end{align*}, which is approximately 99.8%.

### Extension Investigation

Contact a local auto insurance company and ask them on average how often an auto insurance policy holder makes a claim. Then ask them how many people have auto insurance policies with their company. If you don’t have a Ti-84 Calculator, you can use this website: http://www.danielsoper.com/statcalc3/calc.aspx?id=70.

- Then calculate the probability that no more than one personhas an auto insurance policy will make a claim.
- Then calculate the probability that no more than two people have auto insurance policies will make a claim.
- Then calculate the probability that five or less people have auto insurance policies will make a claim.
- Then calculate the probability that half the people that have auto insurance policies will make a claim.
- Analyze your findings. Describe what happens with the probability of making a claim.
- Let’s go back to the original problem, why do you think car insurance is expensive? How do you think car insurance companies determine how much to charge someone for car insurance?

### Resources Cited

http://www.danielsoper.com/statcalc3/calc.aspx?id=70

### Connections to other CK-12 Subject Areas

- Binomial Distribution
- Binomial Distribution and Probability
- Binompdf Function.

Image from: http://www.shutterstock.com/language.en/pic.mhtml?id=13206280&rid=945688