How to Calculate Powerball Lottery Odds

The odds of winning the lottery jackpot or a smaller prize depend on the total number of ticket combinations. The Powerball and Mega Millions lotteries have a similar structure, and if you know a little math, it is easy to calculate the chances of picking the winning ticket.

In the Powerball lottery there are five regular numbers on white balls, and one special number on a red ball, the "Powerball." The five regular numbers must be distinct integers between 1 and 59. The sixth number is between 1 and 35; it does not have to be distinct from the other five since it is drawn from a different set. You win the jackpot if you match all six numbers. In addition, there are various prizes if you match only some of the numbers. Here is a breakdown of the Powerball lottery and the probablility of winning each kind of prize.

All Six Numbers--The Jackpot

The total number of possible Powerball ticket combinations is

  59 x 58 x 57 x 56 x 55
------------------------------ x 35 = 175,223,510
       5 x 4 x 3 x 2 x 1

Therefore, the odds of winning the jackpot are 1 in 175,223,510, or 0.000000005707.

All Five White Balls--$1,000,000

For every winning jackpot ticket, there are 34 other tickets that have the five white balls correct, but not the the powerball. Thus, the odds of winning the $200,000 lottery prize are 34 out of 175,223,510, or 1 out of 5,153,633. This is 0.00000019404.

Four White Balls + The Powerball--$10,000

For every winning jackpot ticket, there are 270 other ticket combinations that have one white ball incorrect. How so? For each of the 5 correct balls, you can replace it with one of 54 incorrect balls. Since 5 x 54 = 270, the odds are 270 in 175,223,510 of winning $10,000. This is nearly equivalent to 1 out of 648,976, or 0.0000015409.

Four White Balls--$100

There are 270 x 34 = 9,180 ways to match only 4 white balls, thus the odds of winning this prize are 9,180 out of 175,223,510. This roughly equivalent to 1 in 19,088, or 0.00005239.

Three White Balls + The Powerball--$100

How many ways can you take 2 of the 5 correct white balls and replace them with 2 out of 54 incorrect white balls? The answer is

 5 x 4       54 x 53
-------- x ------------ = 14,310
 2 x 1         2 x 1

Since 14,310 out of 175,223,510 is approximately 1 in 12,245, the total probability is 0.000081667.

Three White Balls--$7

There are 14,310 x 34 = 486,540 ways to match only 3 white balls. Since 486,540 out of 175,223,510 works out to about 1 in 360, the probability is 0.0027767.

Two White Balls + The Powerball--$7

We must count the number of ways to take 3 out of 5 correct white balls and replace them with 3 out of 54 incorrect balls. This is

 5 x 4 x 3      54 x 53 x 52
------------ x ----------------- = 248,040
 3 x 2 x 1         3 x 2 x 1

The odds are 248,040 out of 175,223,510, or about 1 in 707, or 0.0014156.

One White Ball + The Powerball--$4

The number of ways to get 1 white ball correct and 4 incorrect is

  54 x 53 x 52 x 51
------------------------ x 5 = 1,581,255
      4 x 3 x 2 x 1

Thus the odds of winning the $4 prize are 1,581,255 out of 175,223,510, or about 1 in 111, or 0.0090242.

Just the Powerball--$4

The number of ways to pick 5 incorrect white balls and the correct powerball is

  54 x 53 x 52 x 51 x 50
------------------------------ = 3,162,510
       5 x 4 x 3 x 2 x 1

This is 3,162,510 out of 175,223,510, approximately 1 out of 56, or 0.018048. Notice that the odds are not 1 out of 35. This is because we are counting a more specific quantity: the number of ways of getting the powerball right and the other five balls wrong. If we are computing just the probability of getting the powerball right, and ignoring the white balls, it would be 1 out of 35.

Also note that there is no prize for getting only two white balls correct.

© Had2Know 2010

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