1. 4-4 Probability: The Basis of Insurance
Advanced Financial Algebra

2. How Does Probability Affect Your Auto Insurance?
Insurance companies classify drivers according to their age, sex, marital status, driving record, and locality.
Since insurance companies are businesses that are trying to make a profit, they need to know what types of drivers are more likely to be involved in accidents.
Those drivers are more likely to cost the insurance company money for claims.
A statistician, actuary, uses probability to makes these predictions and the insurance company uses them to determine how much to charge certain people for insurance.
As a teenager, you pay higher rates since teens have proven to be a higher risk.
How Does Probability Affect Your Auto Insurance?

3. Example 1
A pail contains seven red, six green, three orange, and four black marbles. One marble is to be drawn at random. What is the probability that the marble is red?
SOLUTION:
Probability = 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 𝑡𝑜𝑡𝑎𝑙 = # 𝑟𝑒𝑑 𝑚𝑎𝑟𝑏𝑙𝑒𝑠 𝑡𝑜𝑡𝑎𝑙 = 7 𝑟𝑒𝑑 20 𝑡𝑜𝑡𝑎𝑙 ≈ 35%
There is a 35% chance the chose marble is red.

4. Example 2 - question
A factory is planning to rearrange the parking spaces in its parking lot to better serve the employees. The employees work in three different shifts and drive five different types of vehicles, as shown in the two-way table below.
An employee is selected at random. What is the probability that an employee drives an SUV?

5. Example 2 - SOLUTION
Use the Total columns to find the total number of employees, which is 246. Look for the SUV column to find that the total number of SUV drivers is 91. The probability can be found by creating a fraction.
Probability = 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 𝑡𝑜𝑡𝑎𝑙 = # 𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑒𝑠 𝑤ℎ𝑜 𝑑𝑟𝑖𝑣𝑒 𝑆𝑈𝑉𝑠 𝑡𝑜𝑡𝑎𝑙 = 9 𝑆𝑈𝑉 𝑑𝑟𝑖𝑣𝑒𝑟𝑠 246 𝑡𝑜𝑡𝑎𝑙 ≈ 37%
There is a 37% chance that an employee drives an SUV.

6. Example 3
The factory from Example 2 is thinking of closing part of the parking lot at night, since there are fewer night-shift workers. What is the probability that an employee drives a motorcycle, given that they work the night shift?
SOLUTION:
This is an example of conditional probability. Since we are given that the person works the night shift, the denominator will be the total number of people who work the night shift. This is indicated by the yellow cell in the table. The numerator of the fraction is the number of employees who work the night shift and drive a motorcycle. This is indicated by the green cell in the table.

7. Example 3 SOLUTION continued
The letter symbols can be used, along with a vertical line that stands for “given” to create a symbolic shorthand for “the probability the person drives a motorcycle given that they are on the night shift.” This can be written as P(C|N).
If you know that an employee works the night shift, there is an 8% chance that they drive a motorcycle.

8. Example 4 - question
Justin is doing a statistical analysis to check if sports cars in his town are involved in more accidents than other types of cars. He polls 315 randomly selected parents of students in his high school and gets the results shown in the table at right.
Does the table support Justin’s theory that sports cars get into more accidents?

9. Example 4 SOLUTION
Justin needs to examine two probabilities, P(A) and P(A|S). For all types of vehicles in his survey, the probability of getting into an accident is P(A).
If we restrict to the set of sports cars, we can find the conditional probability P(A|S).
Since the probability of having had an accident is the same for the set of sports cars as it is for everyone in the entire survey, Justin did not get support for his theory.
The percentage of sports car drivers who get into accidents is no different than the general percentage of people who got into accidents in Justin’s survey.

10. Independent and Associated Events
Independent Events:
From the previous example #4, we call A and S independent events.
Two events are independent if the probability of one occurring is unaffected by the occurrence of the other event.
If  we say A and B are independent, like they were in example #4.
Associated Events:
If   then we say A and B are associated events.

11. Example 5
Jhanvi is judging a classic car show. She is also preparing a report on the 135 cars in the show. 70 of the cars had a manual transmission, 43 had air conditioning, 13 had both a manual transmission and air conditioning, and 35 had neither. A car will be selected at random to appear on the cover of the report. What is the probability that the car will have air conditioning, but not a manual transmission?
SOLUTION:
Jhanvi uses a Venn diagram to display some of her findings. She lets M represent cars with a manual transmission and A represent cars with air conditioning. She created the Venn diagrams above. Notice that the numbers in each region add up to the total number of cars, 135.
The 30 cars in the section shaded in dark blue below have air conditioning, but do not have manual transmissions.
There is a 22% chance that the car on the cover of the report will have air conditioning but no manual transmission.

12. Example 6
Using this Venn where the letter B is used if the car exterior is black, and T is used if the interior color is tan, what is the conditional probability that a randomly selected car is black given that it has a tan interior?
SOLUTION:
Look at the two distinct regions in the circle representing the tan interior color. Out of the 47 cars with tan interiors, 8 are black, and we use these numbers to form the fraction that represents the probability.
The conditional probability that the car is black given it has a tan interior is approximately 17%.

13. Assignment: pg 239 # 2, 4-6 #2

14. Assignment: pg 239 # 2, 4-6 continued#4

15. Assignment: pg 239 # 2, 4-6 continued#5

16. Assignment: pg 239 # 2, 4-6 continued#6