For example, if you flip a coin, bet $1 that the toss will come up heads, and win $1 if it does come up heads, then the expected value is
On the other hand, if you play a game in which you have a 1 in 100 chance to win, and you win $50 for a $1 bet, your expected value is
What this means is that, on average, you will lose 49 cents each time you bet $1. Some games have more than one way to win, and the expected value calculation is slightly more complicated. For example, suppose you bet $1 on a roll of a die, and you win $2 if 1 is rolled, you win $1 if 2 is rolled, and you lose otherwise. Then the expected value is
which is about 17¢. This means, on average, you will lose about 17¢ each time you bet $1.
In the Pick 3 game you choose a 3-digit number. You then select one of three possible play types, straight, box, or straight/box. When you win, the odds of winning, and how much you win is given by the following table.
| Play Style | Example: | Prize | Odds 1 in: | |
| Your #'s | Numbers Drawn | |||
| Straight (exact order) | 123 | 123 | $500 | 1,000 |
| Box (2 like numbers) | 112 | 112, 121, 211 | $160 | 333.33 |
| Box (3 different numbers) | 123 | 123, 132, 213, 231, 312, 321 |
$80 | 166.67 |
| Straight/Box (2 like #'s)
Exact or Any Order |
112 | 112 | STR $330 | 333.33 |
| 112 | 121, 211 | BOX $80 | ||
| Straight/Box (3 different #'s)
Exact or Any Order |
123 | 123 | STR $290 | 166.67 |
| 123 | 132, 213 231, 312, 321 |
BOX $40 | ||
Problem 1. Verify the odds given in the table for playing straight or box.
Problem 2. Determine the expected value of a $1 bet if you
Which of these games, if any has the best expected value for a person playing Pick 3?
In the Powerball game, you choose 5 (white) numbers from 1 to 53 and 1 (red) Powerball number from 1 to 42. Six winning numbers are chosen, one being the winning Powerball number. For example, here is an example of a set of winning numbers.
There are nine ways to win, depending on how many of the white numbers you
picked and whether or not you picked the winning Powerball number.
| White Balls | Red Balls | Prize | Odds of Winning | Ways to Win |
| 5 | 1 | jackpot | 1:120,526,770 |   |
| 5 | 0 | $100,000 | 1:2,939,677 | |
| 4 | 1 | $5,000 | 1:502,195 | |
| 4 | 0 | $100 | 1:12,249 | |
| 3 | 1 | $100 | 1:10,685 | |
| 3 | 0 | $7 | 1:261 | |
| 2 | 1 | $7 | 1:697 | |
| 1 | 1 | $4 | 1:124 | |
| 0 | 1 | $3 | 1:70 | |
Problem 3. On the back of a Powerball ticket it says that the overall odds of winning a prize are approximately 1:36. Verify whether or not this is a true statement.
Problem 4. There are 1,120,526,770 total different different Powerball ticket. Of these, how many will win the jackpot, and how many will win $100,000?
Problem 5. If the Lottery Commission wishes to have an expected value of -50¢ per $1 bet, calculate what should be the jackpot paid.
Problem 6. If the amount you win was proportional to the odds of winning, calculate what would be the payoffs for each of the ways to win, given that matching 5 white balls and 0 red balls pays $100,000. Round your answers to the nearest dollar.
The Roadrunner Cash game is similar to the Powerball game. In the Roadrunner Cash game you pick 5 numbers out of 31 possible numbers. You win if your numbers include at least three of the winning numbers.
Problem 7. Many times more likely is it to win the top prize in Roadrunner Cash game than it is to win the jackpot in the Powerball game?
Problem 8. Here is another game of chance: Take opaque plastic cups with 3 inch diameter and place them upside down. Place one ball under one cup. A person gets to lift up one cup; they will win if they lift up the cup covering the ball. Suppose you were to line cups, side by side, from California to Maine, a distance of approximately 3000 miles. What is the odds of a person winning this game? How many miles of cups would you need to use in order to play a game which had the same odds of winning as somebody has of winning the jackpot in the Powerball game?