Expected Value

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.

4. Another way to write this: • 1 ½(4) + ½(-2) = 1

5. 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.

6. Example 1 A die is rolled. You receive $1 for each dot that shows. What is the expected value for the game?

7. 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?

8. 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.)

9. 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.

10. 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