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> Coin Statistics

Coin Statistics

Suppose you buy something at the store and get back some coins as change. How many of each type of coin should you expect to get? The pennies are easy to figure … you'll get between zero and four of them, so, two on average. Similarly, on average you'll get 1.5 quarters. Nickels and dimes are more complicated, and have to be considered together. There are five possibilities, as follows.


Adding up the rows and dividing by five shows that the numbers are 0.4 and 0.8, respectively. I've summarized the results in the table below, and also included the expected monetary values of the coins.

coinvalueexpected numberexpected value
penny2   2   ¢
nickel0.42   ¢
dime10¢0.88   ¢

So, among other things, you should expect to get back 4.7 coins on average.

If you want to know more than just the average, it's actually pretty easy to work out the whole distribution of the number of coins. Here it is, if you're curious.


I'll leave the distribution of the value of the coins as an exercise for the reader.

What if you take your coins home and throw them in a jar? Well, once you forget how many sets of coins you've thrown in, you won't be able to calculate the absolute numbers of pennies, nickels, dimes, and quarters; but the relative numbers, or ratios, should be about the same as for a single set, i.e., 20:4:8:15. Similarly, the ratios of the values of the coins should be about 4:4:16:75.

If you wanted to do a bit of measuring or web searching, you could extend the table to include the expected weight and expected volume for each type of coin. Then you'd only need to know how densely the coins pack (tricky!) and how often you get change, and you could figure out how long the jar will take to fill up, and how much it will weigh when it does!

Now, if you take everything I've said so far and consider it as a whole, you have a nice little mathematical model of coin accumulation. It's a good model, very neat and tidy, and I like it a lot … but what I really like is seeing all the different ways it breaks down. It's science to a T … you make a model, test it against reality, see where it fails, and use the failures to refine the model. I don't actually care enough about coins to refine the model, but I do still like seeing the failures. So, here are the ones I've noticed.

First of all, there are different ways to make change. I hope you'll agree that the method I described above is the normal way, but it's certainly not the only way. For example, one could argue that 30¢ ought to be returned as three dimes. The coins are lighter and smaller, and only require reaching into one bin.

That misses the point, though. The question isn't “what's the best way to make change?”, it's “how does this person at the register make change?”; and the answer to that could be almost anything. It's not even constrained by arithmetic … the store might have a “take a penny, leave a penny” thing, or a policy of rounding to the nearest nickel; or the cashier might give you a nickel instead of four pennies, for no reason at all. Or ve might simply make a mistake!

The change you get also depends on what coins and bills are available. If, for example, the cashier is out of dollar bills, ve might buy some from another cashier, or go in back and get more; but ve might also just give you a bunch of quarters instead. A complete science of coins would need to include all that!

Speaking of bills, of course you can work out the same kinds of statistics for bills as for coins. The first three are easy.

billexpected numberexpected value
$  12   $2   
$  50.5$2.5

With twenties and larger bills, though, there are difficulties. Most stores don't stock larger bills, so it doesn't make sense to talk about getting those as change. And, most people don't carry larger bills, so it also doesn't make sense to talk about getting twenties as change … the cashier would just be giving back what the customer had given.

Another difficulty is that the amount of change depends on how you pay, which in turn depends on the size of the purchase. If you're just buying a candy bar or something, you'll probably pay for it with a dollar bill, not a twenty, and then you're guaranteed to get no bills as change. That effect completely dominates any statistics there might be for twenties and larger bills, and skews the statistics even for ones … especially for people (like me) who like to pay $21 instead of $20 on a $10.50 purchase.

By the way, don't confuse yourself like I just did. You don't take your dollar bills home and throw them in a jar, so they don't accumulate, and the average transfer has to be zero; but that doesn't mean the average number you get back is zero, it just means the number you get back matches the number you put in.

In the long run, coins don't accumulate either … the jar fills up and you take it to the bank. Still, it seems reasonable to pretend that they do.

Now, about that effect I was talking about … you can observe the same effect with coins. If something costs less than a dollar, and you pay for it with coins, that skews the statistics; and if something costs more than a dollar but you throw in a few coins anyway to get better change, that skews the statistics too. (Strangely, though, if you always throw in all the coins you have, that doesn't skew the statistics.)

I tend not to carry any coins unless I've already gotten change for something. I imagine that's typical for guys, and that carrying around a coin purse is typical for girls; if so, then you can make an informed guess whether someone is male or female from nothing more than vis coin statistics.

Just to stir things up a little more, here's one last set of thoughts. What if a rogue cashier gives you a 50¢ piece, or one of the various dollar coins? Or a two-dollar bill? Or, perhaps worst of all, a Canadian penny? (Coins and bills are interesting examples to keep in mind for the essay Objects and Identity.)

OK, that wasn't really the last set of thoughts, but it was the last set about how a transaction occurs, given a price. But where do the prices come from? So far I've been pretending the prices are uniformly distributed, but that can't be, because there's no uniform distribution on the positive integers. So, there has to be some distribution of prices, and that distribution will flow through the entire calculation and change the coin statistics.

(It does make some sense to pretend prices are uniform mod $20, and more sense to ignore bills and pretend they're uniform mod $1, but even that is just an approximation.)

I'll be nice and not go into great detail about prices, but here are a few of the factors you'd need to consider. The prices of individual items aren't uniform, because for example they're often set to just less than a dollar multiple … $2.95 or $2.99, say. Sales tax will push that to a small number of cents, the exact number depending on local tax rates. Then you have to take into account the distribution of the number of items you buy, and perhaps correlations between items. Of course this all depends on which items you typically buy at which stores, and how often the stores have sales. There's also the question of where you draw the line between paying cash and using plastic.

Finally, getting back to the original idea of coins in a jar, there are a few things that can perturb the coin statistics directly, without having anything to do with transactions and change. You might use your quarters for laundry, or for parking meters; in the past you might have used your dimes to make phone calls. You might get infected by the “collect them all” meme and hold back some of those commemorative quarters; or you might save a shiny new dime, or some crisp new bills with sequential serial numbers. Coins might roll out of your pocket and get lost in the car seat. (Quarters and dimes seem to do that more than pennies; perhaps you could calculate the probabilities using physics.) And, of course, you might find a penny and pick it up.

Isn't that fun to think about?


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@ November (2006)