IN THIS LESSON, YOU WILL:
Distinguish between theoretical and experimental probability
Explain how the number of trials impacts the accuracy of a prediction
Apply the addition rule
Use the theoretical probability of an event to calculate its expected value
Evaluate and compare strategies using expected values
Use expected value to calculate the value of an insurance policy
Analyze age as a factor that impacts auto insurance premiums
You likely hear about probability in everyday life - maybe looking at a weather prediction, hearing about the risks of texting while driving, or studying the stats on your favorite athlete. But what is probability? Watch the video through 3:34 to learn some basic probability terms, notation, and rules. Then, answer the questions.
After watching the video, complete the missing cells in the table below with either an explanation or example for each vocabulary word.
You flip a fair coin 4 times and get tails once. Which answer correctly identify both the empirical probability and theoretical probability of getting tails on a single coin flip?
Now that you’ve practiced the addition rule, let’s look at a slightly trickier example: what happens when the events are NOT mutually exclusive? Resume the video and answer the questions.
Assume that A and B or not mutually exclusive:
If P(A) = 0.43, P(B) = 0.09, and P(A and B) = .0387, what is P(A or B)?
In a group of 48 students, 30 play video games and 36 have a pet. 27 students both play video games and have a pet. If you randomly choose one student, what is the probability that they play video games or have a pet?
You've been learning about events with uncertain outcomes. Expected value is one tool we can use to make sense of probability and to make decisions when the outcome is uncertain. Expected value is a way to think about all the possible future events using a single number. It includes both the likelihood of each outcome and the value.
Your friend proposes a bet: he’ll roll a fair six-sided die. If it’s an odd number, he’ll give you that many dollars. If it’s an even number, you have to pay him that number of dollars. Should you accept?
Here is the prize wheel. Calculate the expected value of spinning the wheel once.
Because expected value tells you more about the risks and costs you face, it can be helpful when evaluating different insurance options. Read about Hira’s situation below and answer the following questions.
Hira is buying a new phone for $700 and wants to know whether it's worth paying for the additional insurance plan from the phone company. She has the option to add an insurance plan that costs $180 for one year and covers 100% of the cost of repairs.
Hira faces the following risks of damaging her phone over the next year:
P(no damage) = 0.30, costing $0
P(screen damage only) = 0.35, costing $100
P(minor damage) = 0.30, costing $200
P(major damage) = 0.05, costing $500
Note: for simplicity, assume these outcomes are mutually exclusive: only one of them will happen.
What is the probability that Hira has to pay for some kind of phone repair over the next year (either for screen damage, minor damage, or major damage)?
P(screen damage or minor damage or major damage)=?
Hira decides NOT to buy the insurance plan. Instead, she saves $15 per month to put towards phone repairs in case something happens. Do you think that was a good decision? Why or why not?
Age is one of the biggest factors that impacts your car insurance premium. Let’s dive into some data to see how your driving risks and insurance costs change over time.
The table below shows the annual number of crashes that happen in each age group per 100,000 drivers. It includes three different types of crashes and their average costs: crashes that cause property damages, injuries, or fatalities. Use the table to answer the following questions.
Hypothesize: If a law were passed saying that car insurance companies were not allowed to charge different premiums based on age, who do you think would end up paying MORE? Who do you think would end up paying LESS?
Your school is holding a fundraising raffle. They are giving away one prize worth $100 and four prizes worth $25. You buy one ticket for $2 and calculate that you have a 0.00125 chance of winning $100 and a 0.005 chance of winning $25. What is the expected value of your ticket?
E(value of a winning ticket)=
What does your answer from question 24 mean?
Which of the following events are mutually exclusive?
Consider the example from the video where you flip a coin and roll a 6-sided die. Which equation correctly represents the probability of getting a tails or rolling a 6?
Imagine you are picking one card from a standard deck of playing cards, like the one shown below. What is the probability that you pick either a King or a Club? Note: A standard deck of playing cards has 52 total cards. It is split into four different groups of cards, called “suits”: clubs, spades, hearts, and diamonds. Each suit has 13 cards, including the numbers 1-10, one Jack, one Queen, and one King.
Calculate the expected value for the outcomes above. This tells how much Hira should expect to pay in repairs over the next year.
E(repairs for the year)=
How does the expected cost of repairs compare to the cost of the insurance plan?
What is the expected cost of crashes for a driver between 16 and 20 years old?
E(expected cost of crashes for a driver between 16 and 20 years old)=
What is the expected cost of crashes for a driver between 45 and 64 years old?
E(cost of crashes for a driver between 45 and 64 years old)=
Millions of Americans play the lottery every year. Apply your knowledge of probability and expected value to figure out exactly how much a lottery ticket is worth.
What are your chances of winning the Powerball jackpot? Read the rules and answer the questions below.
The Rules of Powerball:
Players select five numbers from 1 to 69 and one Powerball number from 1 to 26
To win $1 million: match the five numbers in any order
To win the jackpot: match the five numbers in any order AND match the Powerball number
There are 11,238,513 different combinations of picking sets of 5 numbers
What is the probability of choosing the 5 winning numbers to win $1 million?
What is the probability of winning the jackpot?
What is the probability of winning $1 million OR winning the jackpot?
Assume the jackpot is currently $350 million and a Powerball ticket costs $2. What is the expected value of a single Powerball ticket?
What is the most you would be willing to pay for a Powerball ticket? Why?
The jackpot value changes each week. How much would the jackpot need to be for a $2 ticket to have a positive expected value?
Your friend Ethan says, “I know your odds of winning the lottery are pretty low. But what if you were really rich? Then, you could buy thousands - or even millions! - of tickets and be pretty much guaranteed to win big.” How would you respond?
