# 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.