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 betAny 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 P x 0

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

ExampleJoe bets $1 on the roll of a dieIf he rolls a 4: he wins $5 plus his bet

2: he wins $1 plus his bet5: he gets his bet back

Is it fair? Ω P O P x O

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 fairIf 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 betAnything else you lose your bet

Ω P O P x O

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 P x 0

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 - 4.99M. E. = -2.82

How can we make this fair?

Changing the M.E.

Joe bets $1 on the roll of a dieIf he rolls a 4: he wins $5 plus his bet

2: he wins $1 plus his bet5: he gets his bet back

Is it fair?Ω P O P x O

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 fairChange an outcome to make this game fair