How to Calculate Powerball Probabilities

- Ph.D., Mathematics, Purdue University
- M.S., Mathematics, Purdue University
- B.A., Mathematics, Physics, and Chemistry, Anderson University

Powerball is a multistate lottery that is quite popular due to its multimillion-dollar jackpots. Some of these jackpots reach values that are well over $100 million. An interesting quest ion from a probabilistic sense is, “How are the odds calculated on the likelihood of winning Powerball?”

### The Rules

First we will examine the rules of Powerball as it is currently configured. During each drawing, two drums full of balls are thoroughly mixed and randomized. The first drum contains white balls numbered 1 to 59. Five are drawn without replacement from this drum. The second drum has red balls that are numbered from 1 to 35. One of these is drawn. The object is to match as many of these numbers as possible.

### The Prizes

The full jackpot is won when all six numbers selected by a player match perfectly with the balls that are drawn. There are prizes with lesser values for partial matching, for a total of nine different ways to win some dollar amount from Powerball. These ways of winning are:

- Matching all five white balls and the red ball wins the grand prize jackpot. The value of this varies depending upon how long it has been since someone has won this grand prize.
- Matching all five white balls but not the red ball wins $1,000,000.
- Matching exactly four of the five white balls and the red ball wins $10,000.
- Matching exactly four of the five white balls but not the red ball wins $100.
- Matching exactly three of the five white balls and the red ball wins $100.
- Matching exactly three of the five white balls but not the red ball wins $7.
- Matching exactly two of the five white balls and the red ball wins $7.
- Matching exactly one of the five white balls and the red ball wins $4.
- Matching just the red ball but none of the white balls wins $4.

We will look at how to calculate each of these probabilities. Throughout these calculations, it is important to note that the order of how the balls come out of the drum is not important. The only thing that matters is the set of balls that are drawn. For this reason our calculations involve combinations and not permutations.

Also useful in every calculation below is the total number of combinations that can be drawn. We have five selected from the 59 white balls, or using the notation for combinations, C(59, 5) = 5,006,386 ways for this to occur. There are 35 ways to select the red ball, resulting in 35 x 5,006,386 = 175,223,510 possible selections.

### Jackpot

Although the jackpot of matching all six balls is the most difficult to obtain, it is the easiest probability to calculate. Out of the multitude of 175,223,510 possible selections, there is exactly one way to win the jackpot. Thus the probability that a particular ticket wins the jackpot is 1/175,223,510.

### Five White Balls

To win $1,000,000 we need to match the five white balls, but not the red one. There is only one way to match all five. There are 34 ways to not match the red ball. So the probability of winning $1,000,000 is 34/175,223,510, or approximately 1/5,153,633.

### Four White Balls and One Red

For a prize of $10,000, we must match four of the five white balls and the red one. There are C(5,4) = 5 ways to match four of the five. The fifth ball must be one of the remaining 54 that were not drawn, and so there are C(54, 1) = 54 ways for this to happen. There is only 1 way to match the red ball. This means that there are 5 x 54 x 1 = 270 ways to match exactly four white balls and the red one, giving a probability of 270/175,223,510, or approximately 1/648,976.

### Four White Balls and No Red

One way to win a prize of $100 is to match four of the five white balls and not match the red one. As in the previous case, there are C(5,4) = 5 ways to match four of the five. The fifth ball must be one of the remaining 54 that were not drawn, and so there are C(54, 1) = 54 ways for this to happen. This time, there are 34 ways to not match the red ball. This means that there are 5 x 54 x 34 = 9180 ways to match exactly four white balls but not the red one, giving a probability of 9180/175,223,510, or approximately 1/19,088.

### Three White Balls and One Red

Another way to win a prize of $100 is to match exactly three of the five white balls and also match the red one. There are C(5,3) = 10 ways to match three of the five. The remaining white balls must be one of the remaining 54 that were not drawn, and so there are C(54, 2) = 1431 ways for this to happen. There is one way to match the red ball. This means that there are 10 x 1431 x 1 = 14,310 ways to match exactly three white balls and the red one, giving a probability of 14,310/175,223,510, or approximately 1/12,245.

### Three White Balls and No Red

One way to win a prize of $7 is to match exactly three of the five white balls and not match the red one. There are C(5,3) = 10 ways to match three of the five. The remaining white balls must be one of the remaining 54 that were not drawn, and so there are C(54, 2) = 1431 ways for this to happen. This time there are 34 ways to not match the red ball. This means that there are 10 x 1431 x 34 = 486,540 ways to match exactly three white balls but not the red one, giving a probability of 486,540/175,223,510, or approximately 1/360.

### Two White Balls and One Red

Another way to win a prize of $7 is to match exactly two of the five white balls and also match the red one. There are C(5,2) = 10 ways to match two of the five. The remaining white balls must be one of the remaining 54 that were not drawn, and so there are C(54, 3) = 24,804 ways for this to happen. There is one way to match the red ball. This means that there are 10 x 24,804 x 1 = 248,040 ways to match exactly two white balls and the red one, giving a probability of 248,040/175,223,510, or approximately 1/706.

### One White Ball and One Red

One way to win a prize of $4 is to match exactly one of the five white balls and also match the red one. There are C(5,4) = 5 ways to match one of the five. The remaining white balls must be one of the remaining 54 that were not drawn, and so there are C(54, 4) = 316,251 ways for this to happen. There is one way to match the red ball. This means that there are 5 x 316,251 x1 = 1,581,255 ways to match exactly one white ball and the red one, giving a probability of 1,581,255/175,223,510, or approximately 1/111.

### One Red Ball

Another way to win a prize of $4 is to match none of the five white balls but match the red one. There are 54 balls that are not any of the five selected, and we have C(54, 5) = 3,162,510 ways for this to happen. There is one way to match the red ball. This means that there are 3,162,510 ways to match none of the balls except for the red one, giving a probability of 3,162,510/175,223,510, or approximately 1/55.

This case is somewhat counterintuitive. There are 36 red balls, so we may think that the probability of matching one of them would be 1/36. However, this neglects the other conditions imposed by the white balls. Many combinations involving the correct red ball also include matches on some of the white balls as well.

See how to calculate the probabilities associated with winning any prize in the Powerball lottery. ]]>