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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