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I’ve been diligently putting all my coins into rollers as soon as I get enough. Some time ago I bought a big bag of coin rollers and have just now run out of quarter rolls. So I wondered to myself, do quarters turn up more often than other denominations?

So I broke out my spreadsheet program and worked it out. I assumed you always get the minimum number of coins (e.g., one quarter and one nickel instead of three dimes), and that each value from $0.00 to $4.99 has equal probability of turning up. This second assumption is the one I’m least sure of, as pricing practices and the effect of taxes may favor an uneven distribution. But that’s a project for another day.

The results are that for any transaction, you should expect to get:

  • 0.8 toonies *
  • 0.4 loonies *
  • 1.5 quarters
  • 0.8 dimes
  • 0.4 nickels
  • 2.0 pennies

I basically used brute force because copy-and-paste is easy, but the same patterns that make copy-and-paste easy mean you don’t really need brute force, and I could probably have done it all in my head. For pennies, you’ll get 1, 2, 3, 4, or 0, and then the pattern repeats. So from just 5 transactions, you can already tell that the average for pennies will be (1+2+3+4+0)/5=2. Loonies are used only for values in the 1 and 3 dollar ranges, and none for 0, 2, and 4, so the average will be (0+1+0+1+0)/5=0.4. The rest are left as an exercise for the reader.

I would have thought I’d run out of penny rollers first. I guess the coin roller people compensated for that but not the rest. When I go buy a new bag I’m going to check what the ratio of denominations is. ‘Cause I’m a dork like that.

* For American folk, replace “loonie” and “toonie” with “one dollar bill” and “the two dollar bill you should have”. With only a one dollar bill, they’re just like pennies—you should expect two every single goddamn time. With a two dollar bill, you’d average just over one bill per transaction. Sure you could use that logic to argue for 3 and 4 dollar bills too, but let’s not get crazy.

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