Honors Stats 4 Day 9 Chapter 16

Do Now Check Your Homework Homework: Chapter 16 p. 382 #1, 2, 4, 5, 6, 17, 18 Objective: SWBAT understand and calculate probability models and expected value Agenda: Do Now Chapter 16 Lesson Practice Worksheet

Do Now Create probability models for the following situations: 1) You get $5 if you roll a 1 or a 6 You get $2 if you roll a 2 or a 3 You get $0 if you roll any other number 2) You spend $1 to play the game If you roll a 6 you get $20 If you roll any other # you get nothing

Probability Models and Expected Value WE WILL BE ABLE TO: Create Probability Models and calculate expected value

- You spend $1 to play the game If you roll a 6 you get $20 You get 2 chances to win If you roll any other # you get nothing (but you spent $1)

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.

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, leaving a 989/1000 chance of neither happening. 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)

An Expected Average The following were the scores for the Chapter 13 Quiz (rounded up): 13, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20 What was the average? What if I wanted to predict the average for next year, what should I do?

An Expected Average What if I wanted to predict the average quiz grade in general for next year knowing that 30% of people got As, 40% of people got Bs, 20% got Cs What should we expect the average quiz grade to be? First, make a probability model

Definitions Probability Model: a table that shows all possible outcomes and their corresponding probabilities Expected value E(x): the average of a data set based on probability Note: sometimes there are endless possibilities! This is called a CONTINUOUS variable. We will be working with DISCRETE variables

Expected Value Formula Expected Value of a Discrete Random Variable =

Create Probability Model, Calculate E(x) 1) a probability model for the 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) a probability model for the amount of money you will win/lose if you buy pops until you win a $20 prize if each pop is $2 (you are willing to spend max $8, once you win you stop buying pop)

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!

Expected Value Let’s go back to our insurance probability model. If you are the insurance company, what should you expect to pay per policy holder (that is, what should you charge each policy holder at a minimum)? X ($)P(x) death250001/1000 disability500010/1000 fine0989/1000

Expected Value Let’s go back to our other Betting Game with the cards. Find the Expected Value Do you think this game is FAIR? In a “FAIR” game E(X)= 0

Expected Value Example Problem Find the expected value given the following probability model for a random variable, x x P(X=x)

Practice Worksheet

Start Homework Chapter 16, page 382 #1, 2, 4, 5, 6, 17, 18 (ignore standard deviation)