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 Probability Value Total Heads Tails

Another way to write this:

Outcomes Probability Value Total Heads ½ 4 ½(4) Tails ½ -2 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 1

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

2 3 4 Outcomes Probability Value Total 1 5 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 Probability Value Total

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.)

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.

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