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## On Coin Distributions

Inspired by a recent foray into Piggy Bank Estimations, I started thinking about the following question:  how are coins distributed?  That is, what percentage of coins in a collection of random change are penniesNickelsDimes?  Quarters?

I began with two assumptions.  They are debatable, like most assumptions are, but they seem like a good place to start an investigation:

1)  Every amount of change is equally likely to be received.

2)  Every amount of change is provided using the minimum number of coins.

What (1) means is that you are just as likely to get 13 cents back in change as you are to get 91 cents when you purchase something.   And (2) means that, when you get that 91 cents back, you’ll get it as 3 quarters, 1 dime, 1 nickel, and 1 penny; not 4 dimes, 9 nickels, and 6 pennies.

I made a chart in Excel of all the possible change amounts from 1 to 99.  I then figured out how many of each coin would be used to provide that amount of change, assuming that change was given efficiently.

Now, assuming each change amount is equally likely, we can simply count the total number of coins and then figure out each percentage as a share of that total.  The total number of coins in the list is 466.  The number of each coin, and it’s approximate percentage, is given below.

By this analysis,  a large, random collection of coins should be roughly 42.5% pennies, 8.5% nickels, 17% dimes, and 32% quarters.   Do me a favor:  the next time you find yourself sitting on a big pile of change, see how it stacks up against these numbers and let me know.

And if you like, you can check this theoretical ratios against the actual numbers in my Piggy Bank.