How to calculate the odds of winning the PowerBall lottery

Although the PowerBall website will give you the odds of winning, this article will explain how to calculate those odds, and more importantly, how small historical changes in the game has greatly decreased your likelihood of winning.

Things You'll Need

• Calculator (or spreadsheet)
• Pen/Paper

Instructions

1. Identify the key elements in calculating the probabilities. For the PowerBall, you're trying to match five numbers from 1-59 (the "white" balls) and one powerball number from 1-39 (the "red" ball).

2. Understand the true probability of winning is the product of matching each ball. i.e. the probabilities of matching the first, second, third, forth, fifth, and powerball (sixth) numbers multiplied out.

3. Understand the probabilities of each number:

   To match the first number, you have 5 chances, because you've chosen five "white" numbers. Each of those numbers can be 1-59, which means you have 59 choices, so your odds will be 5 chances out of 59, or 5/59.

   To match the second number, you only have 4 chances left, since one of your 5 choices was used on ball #1. Likewise, since one of the 59 possible numbers have been used on ball #1, you only have 58 choices left, so your odds are 4 out of 58, or 4/58.

   This continues for balls 3-5, so your odds are 3/57, 2/56, and 1/55, respectively. See next step for more details on these.

   Finally, the powerball is just one chance out of 39 possible numbers, so it's odds will be 1/39.

4. So the odds of matching each number is (rounded):

   1st Number: 5/59 = 0.085
   2nd Number: 4/58 = 0.069
   3rd Number: 3/57 = 0.053
   4rd Number: 2/56 = 0.036
   5th Number: 1/55 = 0.018
   PowerBall: 1/39 = 0.026

5. Now multiply out all those probabilities:
   0.085 x 0.069 x 0.053 x 0.036 x 0.018 x 0.026 = 0.0000000051
   You can present this number in a more meaningful way, be dividing it into 1 (1/0.0000000051), so that the odds of winning the PowerBall are:

   1 in 195,249,054

   (note: those are the true odds, which will differ slightly if you use the rounded figures from above. I rounded them to be less distracting, but for these calculations, you'll want to use the real numbers.)

6. You can simplify this task greatly by using a spreadsheet. In Excel, the formula would simply be:

   =COMBIN(59,5)*COMBIN(39,1)

7. Over the years, the PowerBall has made several changes to the game, usually advertising those changes to "create bigger jackpots". This is certainly true, because your odds of winning dramatically decrease, which results in fewer winners. Below are the changes and odds:

   Apr 22, 1992: 45 white, 45 red = 1 in 54,979,155
   Nov 5, 1997: 49 white, 42 red = 1 in 80,089,128
   Oct 9, 2002: 53 white, 42 red = 1 in 120,526,770
   Aug 28, 2005: 55 white, 42 red = 1 in 146,107,962
   Jan 7, 2009: 59 white, 39 red = 1 in 195,249,054

   So since the PowerBall first started, your odds of winning have gotten over 3.5 times worse. In just the past 7 years, your odds of winning have gotten almost 2.5 times worse!

Resources

• My other articles