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
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
