Mathematical Expectation
A friend offers you the chance to play the
following game:
You bet $2 and roll a die.
If you roll a 6 you win $5 plus your bet
If you roll a 5 you win your bet
Any other number than you roll, you lose
Should you play?
• M.E. is short for mathematical expectation
• M.E. is the amount you will win or lose in a
given situation or experiment.
• M.E. = (probability of winning) x (net gain) +
(probability of losing) x (loss)
* Loss will be a negative number
Is the game fair?
Who is more likely to win?
The game is fair when M.E. = 0
(the odds are 1:1 OR 50:50)
The game is good for the player when M.E. is +
The game is good for the dealer when M.E. is -
We put all the information into a
table
A community organization holds a fundraising
raffle and sells 6000 tickets at $5 each.
• First prize is $10,000
• Second prize is $2000
• Third prize is $1000
• What is the expected value for this lottery?
The Table
Ω
P
O
Px0
1rst place
1/6000 = 0.00017
$9995
1.67
2nd place
1/6000 = 0.00017
$1995
0.33
3rd place
1/6000 = 0.00017
$995
0.17
4th place
5997/6000 = 0.9995
-$5
-4.99
M. E. = 1.67 + 0.33 + 0.17 - 4.99
M. E. = -2.82
The game favors the fundraiser, NOT the ticket holder
How do you know if the game is fair?
Mathematical Expectation = 0
Example
Joe bets $1 on the roll of a die
If he rolls a 4: he wins $5 plus his bet
2: he wins $1 plus his bet
5: he gets his bet back
Is it fair?
Ω
P
O
PxO
4
1/6 = 0.17
5
0.85
2
1/6= 0.17
1
0.17
5
1/6= 0.17
0
0
other
3/6 = 0.5
-1
-0.5
M.E. = 0.85 + 0.17 + 0 – 0.5 = 0.52
This game favors Joe
How do you know if the game is fair?
Mathematical Expectation = 0
If the game favors the player it is NOT fair
If the game favors the dealer it is NOT fair
Example: Is this game fair?
A game costs $4 to play. You roll a die. If you
roll a 1, you keep your bet and win $12.
If you roll a 2 or a 3 you keep your bet
Anything else you lose your bet
Ω
P
O
PxO
1
1/6 = 0.17
12
2
2,3
2/6 = 0.33
0
0
4,5,6
3/6 = 0.5
-4
-2
M.E. = 2 + 0 – 2 = 0
Yes it is fair
6000 tickets at $5 each.
First prize is $10,000 : Second prize is $2000: Third prize is $1000
Ω
P
O
Px0
1rst place
1/6000 = 0.00017
$9995
1.67
2nd place
1/6000 = 0.00017
$1995
0.33
3rd place
1/6000 = 0.00017
$995
0.17
4th place
1/6000 = 0.00017
-$5
-4.99
M. E. = 1.67 + 0.33 + 0.17 - 4.99
M. E. = -2.82
How can we make this fair?
Changing the M.E.
Joe bets $1 on the roll of a die
If he rolls a 4: he wins $5 plus his bet
2: he wins $1 plus his bet
5: he gets his bet back
Is it fair?
Ω
P
O
PxO
4
1/6 = 0.17
5
0.85
2
1/6= 0.17
1
0.17
5
1/6= 0.17
0
0
other
3/6 = 0.5
-1
-0.5
M.E. = 0.85 + 0.17 + 0 – 0.5 = 0.52
Change the bet to make this game fair
Change an outcome to make this game fair