Expected Value Practice Problems
1. Throw a die. If you win $2 when the number is even and lose $1 when the number is odd, what is the
expected value? If you pay $1 to play the game, will you win in the long run? -$0.50
2. A company has a choice of three marketing strategies. The first will cost $150,000 and has a 40% chance of
$1,500,000 in profits and a 60% chance of $500,000 in profits. The second strategy will cost $50,000 and has a
20% chance of $1,000,000 in profits and an 80% chance of $600,000 in profits. The third strategy will cost
$80,000 and has a 50% chance of $1,000,000 in profits and a 50% chance of $400,000 in profits. Which is the
best strategy? first strategy: $750,000; second: 630,000; third: 620,000 – go with first strategy
3. Remove the face cards and the aces from a standard deck, leaving the cards 2 through 10 of each suit.
Choose a card from this smaller deck and look at the number on the card. What is the expected value? 6
4. In a board game, players take turns spinning a wheel with 4 spaces and values of $100, $300, $400, $800.
The probability of landing on $100 is 4/9. The probability of landing on $300 is 2/9. The probability of landing
on $400 is 2/9. The probability of landing on $800 is 1/9. What is the expected value of spinning the wheel
once? What is the expected value of spinning the wheel 5 times? 1 time: $288.89; 5 times: $1444.44
5. An urn contains six balls – three balls are numbered 2, and the others are numbered 5, 6, and 7. You draw
one ball at random. What is the expected value of the number on the ball? 4
6. 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? $5.90
7. 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? 3.5 points
8. There is an equally likely chance that a falling dart will land anywhere on the rug below. The following
system is used to find the number of points the player wins. What is the expected value for the number of
points won? 22 points
Black = 40 points
Gray = 20 points
White = -10 points
9. One hundred fifty tickets for a raffle are sold for $20 per ticket. The 4 winning prizes are $300, $150 and 2
$50 prizes. Find the expected value per ticket. -$16.33
10. A dice game involves rolling 2 dice. If you roll a 2, 3, 4, 10, 11, or a 12 you win $5. If you roll a 5, 6, 7, 8, or
9 you lose $3. Find the expected value you win (or lose) per game. -$0.33
11. Two players choose an integer from 1 to 5. If the product of the two integers is even, then Player A scores
5 points and Player B loses 2 points. If the product of the two integers is odd, then Player B scores 5 points and
Player A loses 2 points. Find the expected value of each player. Player A: 2.48; Player B: 0.52
12. An airline is considering adding a route to the city of New Orleans, Louisiana. Market research predicts that
if the airline serves New Orleans, there is a 42% probability of making a $700,000 profit, a 22% probability of
breaking even, and a 36% probability of losing $1,000,000. What is the expected value of adding a route to
New Orleans? -$66,000
13. A landscaper mows 25 lawns per day on sunny days and 15 lawns per day on cloudy days. If the weather is
sunny 65% and cloudy 35% of the time, how many lawns can he expect to mow per day? 21.5 days
14. You are playing a number cube game where you need 60 points to win. On each turn you roll a pair of dice
(6-sided number cube). If you roll doubles, your score is the product of the numbers. If you do not roll
doubles, you do not score any points. Find the expected value of each turn. How many turns will it take on
average to score 60 points? 2.53 points per game – it will take 23.7, or 24 rolls to get 60 points
15. You pay $3.00 to play. The dealer deals you one card. If it is a spade, you get $10. If it is anything else,
you lose your money. Is this game fair? E(X) = $10 * 13/52 = $10 * 1/4 = $2.50
$2.50 is the return, on average, of the game - - the expected value.
$2.50 < $3.00, so it is not a fair game
16. A casino game costs $3.50 to play. You draw 1 card from a deck. If it is a heart, you win $10; If it is the
Queen of hearts, you win $50. Is this a fair game?
When there are multiple ways to win, the expected value is the sum of how much is won for each probability
as follows.
E(X) = 10(12/52) + 50(1/52) = $2.31 + 0.96 E(X) = $3.27
Since the disco earns 23 cents more, on average, than it pays out to you, it is not a fair game.
17. A player rolls a die and receives the number of dollars equal to the number on the die EXCEPT when the
die shows a 6. If a 6 is rolled, the player loses $6. If the game is to be fair, what should be the cost to play?
E(X) = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5 (1/6) – 6(1/6) = 15/6 – 1 = 9/6
E(X) = $1.50 (3/2 of a dollar)
Charge $1.50 to play
Practice with Expected Value
1. You draw one card from a standard deck of playing cards. If you pick a heart, you will win
$10. If you pick a face card, which is not a heart, you win $8. If you pick any other card, you lose $6. Do you
want to play? Explain.
2. The world famous gambler from Philadelphia, Señor Rick, proposes the following game of
chance. You roll a fair die. If you roll a 1, then Señor Rick pays you $25. If you roll a 2, Señor Rick pays you $5. If
you roll a 3, you win nothing. If you roll a 4 or a 5, you must pay Señor Rick $10, and if you roll a 6, you must
pay Señor Rick $15. Is Señor Rick crazy for proposing such a game? Explain.
3. You pay $10 to play the following game of chance. There is a bag containing 12 balls, five are red, three are
green and the rest are yellow. You are to draw one ball from the bag. You will win $14 if you draw a red ball
and you will win $12 is you draw a yellow ball. How much do you expect to win or loss if you play this game
100 times?
4. A detective figures that he has a one in nine chance of recovering stolen property. His out-ofpocket expenses for the investigation are $9,000. If he is paid his fee only if he recovers the stolen property,
what should he charge clients in order to breakeven?
5. At Tucson Raceway Park, your horse, Soon-to-be-Glue, has a probability of 1/20 of coming in first place, a
probability of 1/10 of coming in second place, and a probability of ¼ of coming in third place. First place pays
$4,500 to the winner, second place $3,500 and third place $1,500. Is it worthwhile to enter the race if it costs
$1,000?
6. Your company plans to invest in a particular project. There is a 35% chance that you will lose $30,000, a 40%
chance that you will break even, and a 25% chance that you will make $55,000. Based solely on this
information, what should you do?
7. A manufacturer is considering the manufacture of a new and better mousetrap. She estimates the
probability that the new mousetrap is successful is 0.75. If it is successful it would generate profits of
$120,000. The development costs for the mousetrap are $98,000. Should the manufacturer proceed with
plans for the new mousetrap? Why or why not?
8. A grab bag contains 12 packages worth 80 cents apiece, 15 packages worth 40 cents each and 25 packages
worth 30 cents apiece. Is it worthwhile to pay 50 cents for the privilege of picking one of the packages at
random?
Answers:
1. Since the expected value of the game is approximately $.42, it is to the player’s advantage
to play the game.
2. Señor Rick is not crazy since the expected value is approximately -.83.
3. You should expect to lose $16.67 after one-hundred games.
4. The detective must charge $81,000.
5. Soon-to-be-Glue is soon to be glue; his net winnings are – $50.
6. The expected value of the project is $3,250. Since the expected value is positive, you should proceed with
the project.
7. The mousetrap will generate profits of 65,500, proceed with the project.
8. Absolutely not – the expected value of your prize is only 23.1 cents.