## 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 **pennies**? **Nickels**? **Dimes? 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.

*Click here to see more in Application.*

I did compare it to your Piggy Bank percentages. Pretty close!