1. Warm Up
What is the type of sampling for 1-3? 1. “The names of 50 contestants are written on 50 cards. The cards are placed in a hat and 10 names are drawn.” 2. A researcher selects 5 males and 5 females out of 50 males and 50 females. 3. What would you have to score to be in the 85th percentile if the average score is a 75 and the standard deviation is 4.7? 4. What would be the minimum and maximum value of flipping a coin when using randInt(

2. Review
Simulation: If Babe Ruth has a 57% chance of hitting a home run every time he is at bat, run a simulation to find out his chances of hitting a homerun 3 times if he get to bat 4 times in a game. Run 15 trials.

3. Day 8: Expected Value
Unit 1: Statistics

4. Today’s Objectives
Students will work with expected value.

5. Expected Value and Fair Games

6. Expected Value
Expected value is the weighted average of all possible outcomes.
For example, if a trial has the outcomes 10, 20 and 60:
The average of 10, 20, and 60 = 30
This assumes an even distribution: 10 20    60

7. Sometimes, outcomes will not have equal likelihoods.X 1 2 3
P(X) .5
.25
E(X) = .5(1) + .25(2) + .25(3) = 1.75

8. You play a game in which you roll one fair die
You play a game in which you roll one fair die. If you roll a 6, you win $5. If you roll a 1 or a 2, you win $2. If you roll anything else, you don’t win any money.
Create a probability model for this game:   6 
1, 2 
3, 4, 5  X 
$5 
$2 
$0 
P(X) 
1/6 
1/3 
1/2 
How much would you be willing to pay to play?
E(X) =$5(1/6) + $2(1/3) + $0(1/2) = $1.50
A price of $1.50 makes this a fair game.

9. E(X) = .05(3500) + .1(2500) + .25(500)+.6(-1000) = -$50.
At Tucson Raceway Park, your horse, My Little Pony, has a probability of 1/20 of coming in first place, a probability of 1/10 of coming in second place, and a probability of ¼ of coming in third place. First place pays $4,500 to the winner, second place $3,500 and third place $1,500. Is it worthwhile to enter the race if it costs $1,000?
1st
2nd
3rd
Other X
$3500
$2500
$500
-$1000
P(X)
.05
.10
.25
.60
E(X) = .05(3500) + .1(2500) + .25(500)+.6(-1000) = -$50.

10. This is the Law of Large Numbers!
What does an expected value of -$50 mean?
Its important to note that nobody will actually lose $50—this is not one of the options. Over a large number of trials, this will be the average loss experienced.
This is the Law of Large Numbers!
Insurance companies and casinos build their businesses based on the law of large numbers.

11. At a carnival there is a game where you get to flip a coin twice
At a carnival there is a game where you get to flip a coin twice. For every head they give you $3, and for every tail you have to pay $1. What should the cost of the game be to make it fair?

12. Questions about expected value?