Recently, my kid had to count our pennies, nickels, dimes, and quarters for homework. It'd be a pointless exercise, except that we had about $100 in change. I pay for every purchase in bills only, and then I take all the change home. Based on this, I wondered what the expected frequencies would be of each coin type, assuming that all change values 0 - 99 are equally likely, and that change is always paid using the fewest coins. Here's how I found those frequencies.
I can get 0, 1, 2, 3, or 4 pennies with equal probability, so I expect 2 pennies.
When I ignore pennies and quarters, the remaining coins must add up to 0, 5, 10, 15, or 20 with equal probability, as follows:
0 = 0 nickels, 0 dimes
5 = 1 nickel, 0 dimes
10 = 0 nickels, 1 dime
15 = 1 nickel, 1 dime
20 = 0 nickels, 2 dimes
That's 2 nickels in 5 scenarios, so I expect 2 / 5 = 0.4 nickels.
And that's 4 dimes in 5 scenarios, so I expect 4 / 5 = 0.8 dimes.
I can get 0, 1, 2, or 3 quarters with equal probability, so I expect 1.5 quarters.
Overall, that's 2 pennies, 0.4 nickels, 0.8 dimes, and 1.5 quarters--a total of 4.7 coins.
I can get any number from 0 to 99 cents in change with equal probability, so I expect 49.5 cents in change. Sure enough, 2 pennies, 0.4 nickels, 0.8 dimes, and 1.5 quarters is worth exactly 49.5 cents.
Of that 49.5 cents, I've got 2 cents in pennies, 2 cents in nickels, 8 cents in dimes, and 37.5 cents in quarters.
Back to my initial problem, about 42.5% of my coins should be pennies, 8.5% nickels, 17% dimes, and 32% quarters.
But most of the value is in quarters. Specifically, about 76% of my change is in quarters, 16% in dimes, 4% in nickels, and 4% in pennies.
No comments:
Post a Comment