Honors Stats 4 Day 10 Chapter 16

SILENT DO NOW ON DESK: Agenda: Ch. 15 Review WS Ch 15 Quiz Prob Book Expected Value DO NOW: Ch 15 Review Homework due Tuesday: Expected Value Worksheet 15 min

Ch. 15 Quiz 25 min

Probability Models 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

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, 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) Create Probability model (5 min)

An Expected Average What if I wanted to predict the average person’s grade in general for next year knowing that 30% of people got As (95s), 40% of people got Bs (85s), 20% got Cs (75s) What should we expect the average grade to be? First, make a probability model

Expected Value Formula Expected Value of a Discrete Random Variable =

Definitions Probability Model: a table that shows all possible outcomes and their corresponding probabilities Expected value E(x): the average of an unknown or future data set based on probability Note: sometimes there are endless possibilities! This is called a CONTINUOUS variable. We will be working with DISCRETE variables 3 min

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) death 25000 1/1000 disability 5000 10/1000 fine 989/1000

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 15 30 40 50 P(X=x) 0.2 0.5 0.1

Standard Deviation A measure of SPREAD The average amount each value (X) deviates from (differs from) the AVERAGE (the AVERAGE when we are not given the data points is… EXPECTED VALUE = E(x))

Remember: Steps to Find SD of the Random Variable Calculate average (E(x)) Find how far each value differs from average (x-E(x)) Calculate the deviations squared to keep them positive, (X-E(X))2 Find average of those deviations: Multiply by their probabilities Add Square Root WE CALCULATE THE AVERAGE DEVIATIONS LIKE THE EXPECTED VALUE BECAUSE ACTUAL DATA POINTS DON’T EXIST

Practice 1 A company makes speakers. One out of every 50 speakers is faulty. The company doesn’t know it is faulty until the customer complains. Suppose the company makes a $6 profit on the sale of any working speaker but suffers a loss of $80 for every faulty speaker unit (they offer a guarantee money back and repair). Find the average profit and standard deviation