Tag Archives: lottery

PowerBall Redux

Donate to a charity instead.  Let me explain.
The majority of responses to my PowerBall description/warnings yesterday have been, “If you don’t play, you can’t win.”  Unfortunately, I know many, many people are buying many lottery tickets, way more than they should.
OK.  For almost everyone, there’s little harm in spending $2 on a ticket for the entertainment, but don’t expect to win, and don’t buy multiple tickets unless you can afford to do without every dollar you spend. I worry about those who are “investing” tens or hundreds of dollars on any lottery.
Two of my school colleagues captured the idea of a lottery yesterday with their analogies,
Steve:  Suppose you go to the beach and grab a handful of sand and bring it back to your house.  And you do that every single day. Then your odds of winning the powerball are still slightly worse than picking out one particular grain of sand from all the sand you accumulated over an entire year.
Or more simply put from the perspective of a lottery official, 
Patrick:  Here’s our idea.  You guys all throw your money in a big pile.  Then, after we take some of it, we’ll give the pile to just one of you.
For perspective, a football field is 120 yards long, or 703.6 US dollars long using the logic of my last post. Rounding up, that would buy you 352 PowerBall tickets. That means investing $704 dollars would buy you a single football field length of chances in 10.5 coast-to-coast traverses of the entire United States.  There’s going to be an incredibly large number of disappointed people tomorrow.
MORAL:  Even an incredibly large multiple of a less-than-microscopic chance is still a less-than-microscopic chance.
BETTER IDEA: Assume you have the resources and are willing to part with tens or hundreds of dollars for no likelihood of tangible personal gain.  Using the $704 football example, buy 2 tickets and donate the other $700 to charity. You’ll do much more good.

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.

Running into Math

Here’s a real-world math problem I just found.

For the last two years, the AJC Peachtree Road Race in Atlanta, the “World’s Largest 10K” (it happens every July 4th), has been using a lottery system to determine which non-invited runners get race numbers.

To accommodate those who would like to participate in the AJC Peachtree Road Race with their family and friends, the lottery registration system allows groups of up to 10 people to enter the lottery as a “Group”.  During the selection process, if a “Group” is selected everyone in the group will receive an entry.  If a “Group” is not selected through the lottery, no one in the group will receive entry into the event.  Those entering the lottery as a group have an equal chance of getting into the event as those entering as individuals (source, emphasis added).

Assume a full group of 10 runners enters as a group.  If any 1 runner in the group is selected in the lottery, every runner in the group gets a race number even if no one else in the group is chosen.  On the surface, this seems like it ought to give a runner a better chance of getting a lottery number if entering as a group.  But … the organizers claim that individuals seeking race numbers have an equal probability of getting into the race whether entering solo or in a group.  So how do they do it?

I didn’t find this problem at the right point in my class’ curriculum sequence this year (I get that I raise lots of rightfully debatable curriculum & teaching issues here), but maybe it will work for one of you.  Even so, I’m trying to create a 10-15 minute gap in an upcoming class to give this problem as a “cool (or real) math moment” that I have from time to time in my courses.  If I can get some student results, I’ll post them here.  I’ll provide links/posts from here to any pages or tweets that tackle this.  Enjoy.