1. Expected Value

2. Take out a coin!
You win 4 dollars for heads, and lose 2 dollars for tails.

3. 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
½(4) + ½(-2) = 1

4. 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.

5. Example 1
A 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

6. Example 2
A $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
Possibility
Probability

7. Example 3
In 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.)

8. 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.

9. Example 4: Solutions
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