### Driving with mathematicians

“How many gas stations do you think we’ll pass on the way home?”

“You mean on the side of the road, or at exits?”

“Whichever - places where we’d consider buying gas.”

“I don’t know, six, eight…why?”

“Well, if we pass the first 1/e gas stations and then stop at the first place that offers cheaper gas, then we’ll maximize the chance that we’ll end up buying the cheapest gas. I mean, assuming that gas prices are random and don’t follow any trend with regards to location, which obviously isn’t the case.”

“Oh. Ha! But that’s not the algorithm we want - that one maximizes the probability of getting the cheapest gas; it doesn’t minimize the expected price of the gas we buy, which is what we want.”

“Oh, yeah, you’re right: if we apply that first algorithm, then in the likely event that we don’t get the very cheapest gas, there’s still a reasonable chance that we’ll end up with pretty expensive gas.”

“Right.”

“So what’s the strategy for minimizing the expected price of gas?”

“I don’t know.”

So, is there a probabilist in the house?