Right on the Money for a Change

By Burkard Polster and Marty Ross

The Age, 18 February 2008

If you are like us then you always have a kilo of loose change rattling around in your pockets. Wouldn’t it be a relief to get rid of it all? Here is a novel scheme for doing just that.

Imagine you are at the checkout in a supermarket and the cashier is going through the usual options with you: Flybuys? Cash or credit? Gamble? Hey, that’s a new one!

If you decide to “gamble” then you would be asked by the cashier to push a special button on the cash register. For example, suppose your bill is $32.56, and so the closest dollar amounts are $32 and $33. Pushing the Gamble Button then produces a random number from 0 to 100, with the exception of 56, and all remaining numbers equally likely. If the number produced is less than 56, then your bill is rounded up to $33, and otherwise you pay $32. In either case you and the cashier only have to exchange bank notes and dollar coins: none of those pocket-filling silver ones.

If you are more adventurous, you could dispense with all the coins by rounding the bill to $30 or $35. In this scenario, a number would be chosen from 0 to 500, with the exception of 256, and you would pay the higher or the smaller amount according to whether your number is lower or higher than 256. Or, you could choose to round to $0 or $100 … However, that may be an overly scary way to buy an ice-cream.

The nifty fact about this scheme is that, although you will win or lose on specific gambles, in the long run there won’t be any difference to you (or the shops). It all evens out.

We have presented this as gambling and a bit of fun, but it is really just simple applied probabilities. And in fact, because there are no longer any 1 or 2 cent coins, you are gambling like this already. Recall that what you pay in a supermarket is not the actual sum of your items, but rather that true sum rounded to the nearest 5 cent multiple. So, each time you pay a little more or a little less, but in the long run it all evens out.

Of course, for our gamble-away-change scheme to function, your decision to push the button would have to be legally binding. Just because $33 comes up as the amount due, you cannot then decide that you’d rather pay the exact amount. Bad luck!

In fact, with the current 1 cent and 2 cent rounding, you can do this and effectively can cheat the system. Simply pay by cash if the sum will be rounded down, and otherwise pay the exact amount by credit card. Spend your 1 or 2 cents profit wisely.