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 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? Outcomes 1 2 3 4 5 6 Probability Value

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

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