Expected number of each coin denomination in change
Alec Jacobson
February 07, 2010
Nickels are rare in change. This is verifiable even in just a experimental and anecdotal way. I look in my change bin at home and there are many quarters, many dimes and pennies and very few nickels. It also makes sense when thinking just cursorily at the likelihoods of each coin denomination popping up in your change at the super market. I was thinking about these likelihoods---more formally the expected amount of each coin after a transaction---a little (too) much and thought I might as well post my findings. So if E[X] is the expected number of times for event X to happen, then I'll write E[pennies] to mean the expected number of pennies in my change after a transaction. I'll assume that the total change is a fair, random number between 0 and 99 cents. I'll also assume that the cashier will give you the minimal amount of coins to make the change.
E[quarters] = 0*25/100 + 1*25/100 + 2*25/100 + 3*25/100 = 1.5
E[dimes] = 0*10/25 + 1*10/25 + 2*5/25 = 0.8
E[nickels] = 0*5/10 + 1*5/10 = 0.5
E[pennies] = 0*1/5 + 1*1/5 + 2*1/5 + 3*1/5 + 4*1/5 = 2
So here's the expected change for a random purchase at the store:
Note: You can even double check the math by noting that the expected total amount in change should be $0.50. And the total amount from the above coins is:
1.5*$0.25 + 0.8*$0.10 + 0.5*$0.05 + 2*$0.01 = $0.50