EXPECTED VALUE

Take out a coin!

You win 4 dollars for heads, and lose 2 dollars for tails.

How could we predict what you would win on average?

Half the time, you’ll win 4 dollars.Half the time, you’ll lose 2 dollars.

Outcomes Heads Tails

Probability

Value

Total

Another way to write this:Outcomes Heads Tails

Probability ½ ½

Value 4 -2

Total ½(4) ½(-2) 1

½(4) + ½(-2) = 1

Expected Value

• Since you’d win $1 on average, it’s the value you could “expect” to win after playing over and over

• Expected Value: The value is what the player can expect to win or lose if they were to play a game many times.

Example 1A die is rolled. You receive $1 for each

dot that shows. What is the expected value for the game?

Outcomes 1 2 3 4 5 6

Probability

Value

Total

Example 2A $20 bill, two $10 bills, three $5

bills and four $1 bills are placed in a bag. If a bill is chosen at random, what is the expected value for the amount chosen?

Outcomes

Probability

Value

Total

Example 3In a game, you flip a coin twice, and record the

number of heads that occur. You get 10 points for 2 heads, zero points for 1 head, and 5 points for no heads. What is the expected value for the number of points you’ll win per turn? (Hint: List every outcome.)

Example 4: Your Turn!Find the expected value (or expectation) of the

games described.• Mike wins $2 if a coin toss shows heads and

$1 if it shows tails.• Jane wins $10 if a die roll shows a six, and she

loses $1 otherwise.• A coin is tossed twice. Albert wins $2 for each

heads and must pay $1 for each tails.

• Mike wins $2 if a coin toss shows heads and $1 if it shows tails– $1.50

• Jane wins $10 if a die roll shows a six, and she loses $1 otherwise– $0.83

• A coin is tossed twice. Albert wins $2 for each heads and must pay $1 for each tails.– $1.00

Example 4: Solutions