Quiz: Calculating the Odds of Winning the Powerball

Posing the Question - What you need to know!

What are the odds of winning the Powerball jackpot? Very small. How small? If you are not into probability and mathematics, please feel free to pass on this section.  Some people just buy a lottery ticket and hope for the best. I am a proud bearer of the nerd flag, so this is right up my alley.

In this little quiz it’s going to be your job to figure the odds. But, don’t worry if you’re a bit rusty  on your math skills!  I’ll walk you through the solution and then you’ll have another chance to redeem yourself in calculating the odds for the second and third tier Powerball prizes. Here are the facts that you need to know:

• A player picks 5 main numbers and 1 Powerball number.
• The main numbers have a range from 1-69.
• The Powerball number has a range from 1-26.
• To win the Powerball jackpot, the player must match all 5 main numbers as well as the Powerball number.
• Hint: Good luck factor(ial)ing the probability!

So what are the odds of winning the Powerball jackpot? Find out the answer below.

Click Here to Show the Answer

Method:

Let’s start by solving for the probability of matching all 5 main numbers.

On the first draw, you have 1 in 69 chances of having the winning number because there are 69 possible numbers to draw from. What happens on the second draw?

On the second draw, you have 1 in 68 chances of having the winning number. Why? Because after the first number is drawn, it is not returned to the pool of numbers being drawn leaving only 68 possible numbers to draw from. Can you guess what happens on the next three draws?

You guessed it! On the next three draws, you have 1 in 67 chances, then 1 in 66 chances, and finally 1 in 65 chances. So, the probability should be 1 in 69*68*67*66*65, right? Not quite. Thankfully, 1 in 1,348,621,560 chances is a far lower probability than what it actually is. So, what are we missing?

The order of the numbers drawn does not make a difference (this dramatically increases our chances of winning, hooray!). In order to make the probability reflect the fact that order does not matter, we need to divide the probability by the number of orders in which they can be drawn. An example of this is that you would win if the numbers are drawn as {1, 2, 3, 4, 5} and also if they were drawn as {3, 2, 5, 4, 1}. So how many orders are there?

For Powerball, since there are 5 main numbers being drawn, there are exactly 5*4*3*2*1 different orders (commonly express as 5 factorial or 5!). So, the last equation for figuring out the probability of matching all of the main numbers in Powerball looks like (69*68*67*66*65)/(5!) which equals 11,238,513. So, you have 1 in 11,238,513 chances of matching all 5 main numbers in the Powerball Lottery.

And then the final step is to take that probability and divide it by the probability of getting the Powerball number for the jackpot (1 in 26). And what you get is 1 in 292,201,338.

For you math geeks at heart, this whole method can be expressed through the combination function:

The n! represents the total number of possible number choices, while the r! represents how many numbers are actually chosen. In Powerball’s case, the n is 69 (the main numbers you can pick from) and then r is 5 (how many main numbers you choose).

The ! is a factorial. With n!, n represents the product of descending numbers starting at n and ending at 1. For example, 5! is 5*4*3*2*1 or 3! is 3*2*1.

This simple equation gives you the probability of matching all 5 main numbers from the Powerball lottery, and then the final step is to once again divide by the probability of matching the winning Powerball number and there you have it! 1 in 292,201,338.

Extra Credit: Second and Third Tier Odds

You have 1 in 292,201,338 chances of winning the Powerball jackpot. But what are the odds of winning the second and even third tier prizes? Leave your answer below in the comments and we’ll let you know if they’re correct!