# PowerBall Math

Given the record size and mania surrounding the current PowerBall Lottery, I thought some of you might be interested in bringing that game into perspective.  This could be an interesting application with some teachers and students.

It certainly is entertaining for many to dream about what you would do if you happened to be lucky enough to win an astronomical lottery.  And lottery vendors are quick to note that your dreams can’t come true if you don’t play.  Nice advertising.  I’ll let the numbers speak to the veracity of the Lottery’s encouragement.

PowerBall is played by picking any 5 different numbers between 1 & 69, and then one PowerBall number between 1 & 26.  So there are $nCr(69,5)*26=292,201,338$ outcomes for this game.  Unfortunately, humans have a particularly difficult time understanding extremely large numbers, so I offer an analogy to bring it a little into perspective.

• The horizontal width of the United States is generally reported to be 2680 miles, and a U.S. dollar bill is 6.14 inches wide.  That means the U.S. is approximately 27,655,505 dollar bills wide.
• If I have 292,201,338 dollar bills (one for every possible PowerBall outcome), I could make a line of dollar bills placed end-to-end from the U.S. East Coast all the way to the West Coast, back to the East, back to the West, and so forth, passing back and forth between the two coasts just over 10.5 times.
• Now imagine that exactly one of those dollar bills was replaced with a replica dollar bill made from gold colored paper.

Your chances of winning the PowerBall lottery are the same as randomly selecting that single gold note from all of those dollar bills laid end-to-end and crossing the entire breadth of the United States 10.5 times.

Dreaming is fun, but how likely is this particular dream to become real?

Play the lottery if doing so is entertaining to you, but like going to the movie theater, don’t expect to get any money back in return.

### 2 responses to “PowerBall Math”

1. This is a really lovely model for hammering home the unlikelihood of winning this particular lottery. Thanks much for providing it. Of course, anyone who has taught probability as far as the topic of expected value is likely to have tried to help students see how poor a return such things are on the money invested, paling in comparison with casino odds (which are still a losing proposition in the long haul, but much, much more slowly on a dollar for dollar basis). Invariably, some people will say (or think), “Sure, but somebody has to win. Why not me?” And the lottery promoters fuel things with slogans like, “You have to be in it to win it.” Seems to me that for all intents and purposes, you win precisely by NOT being in it.

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