1 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Lesson Objectives  Understand the meaning of “expected value.” (Know what it is NOT!)  Know what is meant by “a fair game.”  Find the Expected Value for discrete random variables.  Find the variance and standard deviation for discrete random variables.  Know how to use these values for making decisions.

2 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Expected Value If we repeat an experiment many, many times under the same conditions, and if we average the results, then this average is called the expected value. We call it , or “the mean”

3 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 “Law of Large Numbers” Probability = Long-run relative frequency Expected = Long-run Value average

4 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Learn how to leave a Casino with more money than you ever thought you could leave with!!!

5 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Properties for Discrete r.v < p i < 1, for each i 2. Sum of all probabilities must equal ________.

6 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Mean of a Discrete R.V. p i = P( X = x i )  x =  x i  p i = x 1  p 1 + x 2  p x k  p k

7 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Roll one die many, many times. What do you expect the average to be? xixi pipi /6 1/6 1/6  x = 1  (1/6) + 2  (1/6)  (1/6) =

8 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 A “Fair Bet”: one in which the expected value of the winnings is zero How much would you be willing to pay to play?

9 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002  Don’t pay any more than this to play. Payoff $s Probability3/6 2/6 1/  x =

10 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Variance of a Discrete R.V.  2 =  ( x i -  X ) 2  p i Variance is a measure of RISK. Standard Deviation:  =  2

11 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 The “Wheel” problem: Payoff $s Probability3/6 2/6 1/  2 =

12 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Find the mean and variance for the following discrete r.v. Payoff $s Probability1/2 1/3 1/  2 =  x =  =

13 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Bigger variance More Risk

14 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Which r.v. has the greater risk? XiXi pipi YiYi pipi  x =  y =  2 =  =

15 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Roulette Wheel 18 Red, 18 Black, 2 Green 18 odd 18 even 12 low 12 middle 12 high 12 low 12 middle 12 high 38 slots:

16 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Roulette to 2419 to 361 to to 361 to 121 to 1813 to to 36 EVEN ODD 0 00 FIRST COL. SEC COL. THIRD COL. RED BLACK

17 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Bet $10 on Red many, many times. Let X = Net amount won per play:

18 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Bet $10 on Red many, many times. Let X = Net amount won per play: XiXi pipi In the long run, you will lose an average of $____ per spin of the wheel.

19 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Bet $10 on #12 many, many times. Let X = Net amount won per play:

20 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Bet $10 on #12 many, many times. Let X = Net amount won per play: XiXi pipi  x = In the long run, you will lose an average of $____ per spin of the wheel.

21 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Strategy xx 22 $10 on Red $10 on #12 Summary:

22 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Simulated Roulette Play 1000 individual plays. A one dollar bet on “Red” and a one dollar bet on “12”.

23 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002

24 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002

25 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002

26 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002

27 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Simulated Roulette Play Results for 500 players, each playing 1000 times. J16_Simul means roulette.doc used in MINITAB on the Command Line.

28 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002