Expected Value

Today we're at the county fair. We've got a giant turkey leg, an even bigger funnel cake, and a mission. This year, we're not just here to ride the rides and see the sights. We need to get our Grandma a gift. We may have forgotten until this morning that she's coming for a visit tomorrow. Whoops.

Anyway, we found the perfect gift—a big, stuffed SpongeBob doll. Grandma loves her some SpongeBob. The problem is, we're a few bucks short. As luck would have it, there's a booth right next door that says ,"Need a few bucks? Step on up." It's a dice rolling game, where you pay $5 to roll the die, and you win some cash on rolling a 5 or 6.

We've been feeling real lucky today. In hindsight, though, that was probably us feeling full from all the turkey we ate, because we lost all our money. That was not supposed to happen. We asked our lucky rabbit, an Ouija board, and our horoscope, and they all said to go for it. We should have listened to Grandma instead: she always says to calculate the expected value before playing any game of chance.

Expecto Pendo, Please

Think of the expected value as a long-term average. For instance, when rolling a single die, the die roll can be a 1, a 6, or something in between. If we roll a lot of dice, though, and record the results each time, we can calculate an average die result. Do we tend to roll high? Do we tend to roll low? Do we have too much time on our hands?

Not really, because we can calculate the expected value without rattling all those bones. Here's the formula:

EV = x₁p₁ + x₂p₂ + … + x_np_n

Or, if you like summation notation more, we can write this as:

So, uh, what are all those x's and p's for? Each x is an outcome from whatever we're interested in. In this case, rolling a 1, 2, 3, 4, 5, or 6 are the six possible outcomes for rolling a die. The p's are the probability for each outcome to occur. To find the expected value, we multiply each value by how likely it is to occur, and then add them all together.

= 3.5

While our individual die rolls will vary a lot, on average we'll get a result of 3.5 over the long haul. We can't actually roll a 3.5, but that's okay, because we're averaging the results of multiple rolls. We needed to roll a 5 or 6 to win any money at that game of chance, so this helps to explain why we lost it all.

We can go one step further, though, and find the expected value for how much money, on average, we'll win each time we play the game. Or how much we'll lose, as the case may be. Here are the details for how much we win for every die roll.

We only make money on a 5 or 6, and we get a fistful of nothin' otherwise. However, we can't forget that we paid $5 every time we played, so that has to be taken into account.

Now we can find the expected value. Our outcomes are the amount of money that changes hands. The probabilities are the same as last time. C'mon, big money, no whammies.

= -$0.83

That counts as a whammy, we think. On average, anyone who plays this game is going to lose money. Sure, someone might get lucky and get $15 on their first roll. But eventually things will even out, and the guy running the booth will get his cut in the end.

That makes a lot of sense, actually. People don't go to the county fair to just give their money away to people. Except we just kind of did that. Maybe we should have listened to Grandma before this happened.