UNR, MATH/STAT 352, Spring 2007

UNR, MATH/STAT 352, Spring 2007

UNR, MATH/STAT 352, Spring 2007

Head

Tail

Tossing a symmetric coin

You are paying $1

How much should you getto make the game fair?

UNR, MATH/STAT 352, Spring 2007

Head

Tail

Tossing a symmetric coin

You are paying $1

You are getting $1

Fair g

am

e

UNR, MATH/STAT 352, Spring 2007

Is a randomly chosen cardthe Queen of Spades?

How much should you getto make the game fair?

You pay $1

UNR, MATH/STAT 352, Spring 2007

Is a randomly chosen cardthe Queen of Spades?

P(Q♠) = 1/52 ≈ 0.02

UNR, MATH/STAT 352, Spring 2007

Is a randomly chosen cardthe Queen of Spades?

You get $51

You pay $1

Fair g

am

e

UNR, MATH/STAT 352, Spring 2007

UNR, MATH/STAT 352, Spring 2007

In statistics, a fair game is a game with zero expected payoff.

That is, let random variable X represent the amount that a player gets after one game; X > 0 means profit, X<0 means loss.

1 1

2 2

, with probability

, with probability

...

, with probability n n

S p

S pX

S p

Fair game means that 1 1( ) ... 0n nE X S p S p

UNR, MATH/STAT 352, Spring 2007

In statistics, a fair game is a game with zero expected payoff.

In the coin tossing game, we have

1

2 1

$ , 1/ 2

-$1, 1 1/ 2

x pX

p p

1 1( ) 1 0

2 21

1 2 12

E X x

x

UNR, MATH/STAT 352, Spring 2007

In statistics, a fair game is a game with zero expected payoff.

In the “Queen of spades” game, we have

1

2 1

$ , 1/ 52

-$1, 1 51/ 52

x pX

p p

1 51( ) 1 0

52 5251

1 52 5152

E X x

x

UNR, MATH/STAT 352, Spring 2007

In the “Deal or no Deal” game, we have

1 1 1 1( ) 25 75,000 400,000 750,000 306,256.25

4 4 4 4E X

1

2

3

4

1$25, 41$75,000, 41$400,000, 41$750,000, 4

p

pX

p

p