Homework Time!

At this point, what are the odds of getting two quarters with the same state depiction in your change?  No really…somebody do the math!

If only there was somebody around here who was good at math.  You know...like an engineer or something.  

So while things like statistical process control and giant spreadsheets of calculations are an important and almost definitive part of my job, I'm not so much all about probabilities and stuff like that.  I don't apologize for that.  I took a statistics class back in college, and that dealt with this type of thing, but it was a few years ago, it was really diverse, was taught by a guy more interested in selling his own textbook than teaching the course, and spent more time dealing with gradients and curls than probabilities. 

Also, this probability question is like...really hard.  

Here's why it's difficult.  You certainly only have a finite number of quarters in your sample to choose from...because there are only a certain number in circulation.  Factor in the different types of quarters...there's the old-school ones with the eagle on the back, plus 50 states, a couple territories, and now some national parks.  There isn't an equal number of any of these variations...so you have to take into account some sort of distribution having to do with the year the design was minted, the number of each that were made, the number remaining in circulation after collectors, random discards, and those coin-pressing machines at amusement parks.  Now you're left with a percentage of circulating quarters that contains a specific state design on the back.  

Now, deal with the composition of your change.  If we assume that every 1-cent increment of change from 1 to 99 cents occurs evenly (I doubt this is the case, but much like the Spherical Chicken In A Vacuum, let's just run with it.), you effectively have 50% of the time where getting two matched quarters is impossible because you don't get two quarters.  That is, anything between 1 and 49 cents, would only contain at most one quarter.  (This assumes, of course, that your Cashier Is Familiar With Money, which is sadly, no longer a valid assumption.)  25% of the time, you have one shot at getting it right, as anything between 50 and 74 cents will contain two quarters which would have to match.  From 75 through 99 cents, you get three quarters, and since only two of them have to match each other, and we don't care which two, your odds of meeting your criteria are dramatically increased. 

So all of this came about because I got 90 cents change from the Adorably Tiny Cafeteria Thing In My Building and two of the quarters were 1999 quarters with Connecticut on the back.  I thought to myself, "Hmm...what are the odds of that?"  Now, I actually want to know what the odds are, and I don't really feel like going through all the nonsense necessary to get the answer.  It's a pretty dark corner I've painted myself into.