1M14 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.

2M14 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, thenthis average is called theexpected value.

We call it , or “the mean”

3M14 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

4M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002

Learn how to Learn how to

leave a Casinoleave a Casino

with more moneywith more money

than you ever thought

than you ever thought

you could leave with!!!

you could leave with!!!

Learn how to Learn how to

leave a Casinoleave a Casino

with more moneywith more money

than you ever thought

than you ever thought

you could leave with!!!

you could leave with!!!

5M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002

Properties for Discrete r.v.1. 0 < pi < 1, for each i

2. Sum of all probabilities

must equal ________.

6M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002

Mean of a Discrete R.V.

pi = P( X = xi )

x = xi pi

= x1p1 + x2p2 +...+ xkpk

7M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002

Roll one die many, many times.What do you expect the averageto be?

xi

pi

1 2 3 4 5 6

1/6 1/6 1/6 1/6 1/6 1/6

x = 1(1/6) + 2(1/6) + ... + 6(1/6)

=

8M14 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.

1

2 3

How much wouldyou be willingto pay to play?

How much wouldyou be willingto pay to play?

9M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002

Don’t pay any more than this to play.

Payoff $s

Probability 3/6 2/6 1/6

1 2 3

x =

10M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002

Variance of a Discrete R.V.

2 = (xi - X)2 pi

Variance is a measure of RISK.

Standard Deviation: = 2

11M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002

The “Wheel” problem:

Payoff $s

Probability 3/6 2/6 1/6

1 2 3

2 =

12M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002

Find the mean and variancefor the following discrete r.v.

Payoff $s

Probability 1/2 1/3 1/6

1 2 5

2 =x =

=

13M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002

Bigger Bigger variancevariance

MoreMoreRiskRisk

14M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002

Which r.v. has the greater risk?

Xi

pi

1.9 2.1

.5 .5

Yi

pi

-100 104

.5 .5

x =

y =

2 =

2 =

=

=

15M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002

Roulette Wheel

18 Red, 18 Black, 2 Green

18 Red, 18 Black, 2 Green

18 odd18 even18 odd18 even

12 low12 middle12 high

12 low12 middle12 high

38 slots:

16M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002

RoulettRoulettee

12

3

45

6

78

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

32

31

33

34

35

36

13 to 2419 to 36 1 to 1225 to 361 to 121 to 18 13 to 24

25 to 36 EVENODD

000

FIR

STC

OL.

SEC

CO

L.TH

IRD

CO

L.

REDBLACK

17M14 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:

18M14 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:

Xi

pi

+10 -10

In the long run, you will lose anaverage of $____ per spin of the wheel.

19M14 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:

20M14 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:

Xi

pi

+350 -10

x =

In the long run, you will lose anaverage of $____ per spin of the wheel.

21M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002

Strategy x 2

$10 on Red

$10 on #12

Summary:

22M14 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”.

23M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002200 400 600 800 1000

-250

-200

-150

-100

-50

0

Play Number

Win

ning

sBet on or ?"Red" "#12"

Standard DeviationsEach Play, 1000 Plays

.9986 31.575.7626 182.23

Mean = -52.63

24M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002200 400 600 800 1000

-100

0

100

200

Play Number

Win

ning

sBet on or ?"Red" "#12"

Standard DeviationsEach Play, 1000 Plays

.9986 31.575.7626 182.23

Mean = -52.63

25M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002200 400 600 800 1000

-200

-100

0

100

200

Play Number

Win

ning

sBet on or ?"Red" "#12"

Standard DeviationsEach Play, 1000 Plays

.9986 31.575.7626 182.23

Mean = -52.63

26M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002200 400 600 800 1000

-100

0

100

200

300

400

500

Play Number

Win

ning

sBet on or ?"Red" "#12"

Standard DeviationsEach Play, 1000 Plays

.9986 31.575.7626 182.23

Mean = -52.63

27M14 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.

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

-800 -600 -400 -200 0 200 400 600 800

0

40

80

120

160

200

Ran 12

Fre

quen

cy

Betting on 'Red'500 people, 1000 bets each

-800 -600 -400 -200 0 200 400 600 8000

40

80

120

Winnings

Fre

qu

en

cy

Betting on '12'500 people, 1000 bets each