Stats 4 Day 20b

SILENT DO NOW ON DESK: Agenda: Notebook Ch. 16 Notes HW Time DO NOW: ½ sheet on averages Homework due Monday: Expected Value Worksheet 10 min

Objective SWBAT calculate expected value and define it as an average value.

What is a probability model? Probability Model: a table that shows all possible outcomes and their corresponding probabilities Note: sometimes there are endless possibilities! This is called a CONTINUOUS variable. We will be working with DISCRETE variables 3 min

Expected Value Take the average of this data set of test scores: 62, 88, 90, 85, 74, 78, 78, 80, 85, 65 If I told you that 15% of students get D’s (say 65), 30% get C’s (75), 40% get Bs (85) and 15% get A’s (95), what would you expect the average to be? **Note: make a probability model ***Note: the 3rd column is like saying “how many out of 100” will get that grade

Expected Value Formula and Idea Expected Value of a Discrete Random Variable = The average value of an unknown or future data set- the expected average of repeated trials.

Expected Value You roll a die – you win $24 if you get a 6, $12 for a 1 or 2, and nothing for everything else. How much would you be willing to pay to play this game? Let’s test it out!

Insurance How does insurance work? You pay a yearly deductible- a certain amount that you must give the insurance company each year in order to be covered in case of expenses (ex: break your leg). How do they determine the amount of your deductible? They use probability models and expected value! Also consider life insurance… You pay a monthly fee for a given payout in case of death, so if you die within the time period that you are covered for, the insurance company must pay the agreed upon amount. They must consider the probability of death. Create probability model (5 min)

Example An insurance company decides to give a $25,000 payout for death, $5000 payout for disability, and $0 for neither. If there is a 1/1000 chance of death for someone your age in your area and a 10/1000 chance of disability, create a probability model displaying the possible outcomes.

Betting Games Consider this complicated example using cards (52 cards in a deck)… I will propose this game to you: You pay $5 to play this game. If you pull the ace of hearts, I will give you $100. If you pull any of the other 3 aces, I will give you $10. If you pull any other of the 12 hearts, I will give you your $5 back. If you pull any of the other 36 non-heart cards, you get nothing and I keep your $5. Create a probability model for your EARNINGS (remember, you give me $5 at the beginning. Represent a loss with a negative) Create Probability model (5 min)

Expected Value 1. Set up Probability Model 2. Multiply x times P(x) – multiply across 3. Add all the xP(x) – add down

Expected Value Example Problem Find the expected value given the following probability model for a random variable, x x 15 30 40 50 P(X=x) 0.2 0.5 0.1

Expected Value Let’s go back to our Betting Game with the cards from last class: You pay $5 to play this game. If you pull the ace of hearts, I will give you $100. If you pull any of the other 3 aces, I will give you $10. If you pull any other of the 12 hearts, I will give you your $5 back. If you pull any of the other 36 non-heart cards, you get nothing and I keep your $5. Find the Expected Value Do you think this game is FAIR?

“Fair” A completely “fair” model is one in which the expected value, E(X), equals… E(X)=0

Create Probability Model, Calculate E(x) 1) find the average number of pops you have to buy until winning a prize when there is a 10% chance of winning (you will buy max 4) 2) find the average amount of money you will win/lose if you buy pops until you win a $20 prize if each pop is $2 and the probability of winning is still 10% (you are willing to spend max $8, once you win you stop buying pop)

Expected Value Let’s go back to insurance from last class. An insurance company decides to give a $25,000 payout for death, $5000 payout for disability, and $0 for neither. If there is a 1/1000 chance of death for someone your age in your area and a 10/1000 chance of disability, create a probability model displaying the possible outcomes. If you are the insurance company, what should you expect to pay per policy holder? Knowing this, if you want to make $100 on each customer, what should you charge each policy holder?

Worksheet 1. Create a probability model 2. Calculate the expected value Note: When placing bets or playing a game, if E(X)>0, then you should play! Expected value ws1