Here’s a cute little observation from a while ago that I thought might interest some people. I thought it’d write it up now because frankly, now that the semester is starting, I probably won’t have time to write more substantial posts.

Anyway, maybe this is just me, but I’ve always wondered what’s stopping people from mixing more expensive and less expensive produce in the same bag in the grocery store. I happen to really like apples, so for the sake of concreteness, say Honeycrisp apples are much more expensive than Fuji apples. What’s to stop someone from trying to sneak a couple of Honeycrisps in with a bag of Fuji apples?^{1} Usually, the cashier only checks one of the apples for the price.

If you think about it for a moment, the answer is obvious: you expect to pay the same price anyway! Suppose that your apples have prices

$\{ a_1, a_2, a_3, \ldots, a_n \},$

so that you *should* pay

$\sum_{i=1}^{n} a_i.$

If the cashier picks an apple uniformly at random, the expected amount you pay is just

$\sum_{i=1}^{n} \left(n \cdot a_i\right) \cdot \frac{1}{n} = \sum_{i=1}^{n} a_i.$

since you pay $(n \cdot a_i)$ with probability $1/n$. Of course, you can try to influence this by arranging the apples so that the cashier is more likely to pick a cheap one (e.g. by placing the cheap ones on top), or even confessing to the scheme if you are “caught” and asked to pay more than the actual total value, but this is a pretty scummy thing to do.

I mean, except for the fact that it’s not ethical.↩