Name: _______________________________________
Date:
_______________________________________
Teacher: _______________________________________
Expected Payoff for a Game of Chance
Web resource:
http://www.math.uah.edu/stat/games/index.html
Games of chance include card games, lotteries, some television game shows, raffles, and much
more. In general, a game of chance is any game where chance, rather than skill, determines the
outcome. The outcome is influenced by a randomizing device, and some type of wager, not
necessarily monetary, is made. Some examples of randomizing devices are six-sided number
cubes, cards, spinners, drawings, and coins.
Some games of chance are fair, meaning nobody has an advantage or is expected to win more
than another player or the person running the game. Other games are designed to give an
advantage to the person running the game. By calculating the expected payoff of a game, you
can determine how much you can expect to win or lose on average each time you play the
game.
Example:
At a carnival, you are asked to play a game. For $10 you get to roll a fair six sided
number cube twice. If the number cube lands on an even number once, you win
$8. If the number cube lands on an even number twice, you win $18. Over time,
how much will you win or lose on average each time you play the game?
A six-sided number cube has three even numbers and three odd numbers, so the chance of
rolling an even number on any given roll is
. There are four equally possible outcomes,
each with a probability of : even-even, even-odd, odd-even, odd-odd.
Since there are two ways to roll one even, either even-odd or odd-even, half the time you will
win $8. This is a loss of $2 for you. The probability of rolling an even number twice is . If you
do, you win $18, which is a gain of $8 for you. You also have a chance of not rolling any even
numbers and winning nothing. This would be a loss of $10 for you.
Based on this information, the expected payoff is:
( )
(half the time you will lose $2)
( )
(
(a quarter of the time you will win $8)
)
(a quarter of the time you will lose $10)
Combining these, we get –1 + 2 + –2.5 = –1.5. On average, you will lose $1.50 each time you
play the game. This does not mean that you will never win money, just that over time you will
lose about $1.50 per game.
_____________________________________________________________________________________
© 2012 CompassLearning, Inc.
A2928
Name: _______________________________________
Date:
_______________________________________
Teacher: _______________________________________
Practice:
1) A friend asks you to play a game using a fair six-sided number cube. If you roll a
2 you win 4 points, and if you roll a 5 you win 7 points. Rolling any other number
causes you to lose 3 points. What is the expected payoff? If the game is not fair,
who has the advantage? Explain your reasoning.
2) At a carnival you see a game with the spinner pictured below. The game costs 1
token per spin. If you land on white, you win 3 tokens. If you land on red or
green, you win 1 token. If you land on blue, you lose. What is the expected
payoff? If the game is not fair, who has the advantage? Explain your reasoning.
3) A card game using a standard deck of 52 cards costs $1 to play. If you draw a
face card or an even number, you lose. If you draw an ace, you win $5, and if
you draw an odd number, you win $2. What is the expected payoff? If the game
is not fair, who has the advantage? Explain your reasoning.
_____________________________________________________________________________________
© 2012 CompassLearning, Inc.
A2928