The vending machine or the spring?

Zereen, a thirsty traveler, has come to a fork in the road. If she goes left, she thinks she has a 3-1 chance of finding a vending machine. (Maybe there’s an old sign indicating the existence of such a machine.) However, this vending machine has a downside: The drinks aren’t cooled at all. On the other hand, this vending machine doesn’t require money! Just press the button corresponding to your desired drink, and a warm beverage will appear. If she doesn’t find a vending machine, she reckons she’ll find nothing of value.

If she goes right, she thinks she has a 5-1 chance of coming to a cool, clear spring, and a 1-5 chance of finding nothing of value.

Which path should she choose?

To answer, we need more information. Specifically, we need to know

  1. how much Zereen values the vending machine, and
  2. how much she values spring water.

If we know those details, we can perform a calculation that will yield the correct answer.

So, without further ado, here are the details. In her current situation, Zereen is willing to pay $3 for a can of warm cola, and $2 for a canteen of cold water.

Now we’re in a position to find, and then compare, the expected value of these two choices.

It’s pretty straightforward. You just find the value of each outcome, and then multiply it by the probability of that outcome, and then add all the results together. (You can do this in terms of odds instead of probabilities if you want to, but I’m sticking with probabilities for today. Odds will become important later, when I talk about betting.)

Zereen’s first choice is to go left. There are two possible outcomes: a vending machine, valued at $3, and nothing, valued at 0. The probability of finding the vending machine is 3/4 = .75, and .75 x $3 = $2.25. The probability of finding nothing is .25, and .25 x 0 = 0 (surprise!). Now add them together: $2.25 + 0 = $2.25. That’s the expected value of going left.*

Zereen’s second choice is to go right. Again, there are two possible outcomes: a spring valued at $2, and nothing, valued at 0. The probability of finding the spring is 5/6, and 5/6 x $2 = $1.67. Just as above, the “nothing” outcome adds nothing to our expected value, so the expected value of going right is $1.67.

Since the value is higher for going left than it is for going right, Zereen should go left.

So, to recap: To find out which to choose, we needed to know the possible outcomes, the value of each outcome, and the probability of each outcome. Equipped with this information, we were able to determine Zereen’s optimal choice.

For more information, see Wikipedia on expected utility.

*Relative to staying put, that is. (If this were the expected value relative to going right, then since it’s positive, her choice would already be settled: Go left, young woman!)