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UNR, MATH/STAT 352, Spring 2007
UNR, MATH/STAT 352, Spring 2007
Tail
Head
Tossing a symmetric coin
You are paying $1
How much should you get
to make the game fair?
UNR, MATH/STAT 352, Spring 2007
Head
You are getting $1
UNR, MATH/STAT 352, Spring 2007
Fair game
You are paying $1
Tail
Tossing a symmetric coin
Is a randomly chosen card
the Queen of Spades?
How much should you get
to make the game fair?
You pay $1
UNR, MATH/STAT 352, Spring 2007
Is a randomly chosen card
the Queen of Spades?
P(Q♠) = 1/52 ≈ 0.02
UNR, MATH/STAT 352, Spring 2007
Is a randomly chosen card
the Queen of Spades?
You pay $1
UNR, MATH/STAT 352, Spring 2007
Fair game
You get $51
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.
 S1 , with probability p1
 S , with probability p

2
X  2
...

 Sn , with probability pn
Fair game means that
E ( X )  S1 p1  ...  Sn pn  0
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
$ x, p1  1/ 2

X 
-$1, p2  1  p1  1/ 2
1
1
E( X )  x   1  0
2
2
1
x  1  2  1
2
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
$ x, p1  1/ 52

X 
-$1, p2  1  p1  51/ 52
1
51
E( X )  x   1
0
52
52
51
x  1   52  51
52
UNR, MATH/STAT 352, Spring 2007
In the “Deal or no Deal” game, we have
 $25, p1  1
4

 $75,000, p2  1

4
X 
$400,000, p3  1 4

$750,000, p4  1 4
1
1
1
1
E ( X )  25   75,000   400,000   750,000   306, 256.25
4
4
4
4
UNR, MATH/STAT 352, Spring 2007