Expected Value

I mentioned the idea of “expected value” in my post last week. I believe that the concept of expected value is one of the core principles that is critical to success, not only in the world of Wall Street, but in all areas of life. I talked the other day about the ideas of risk and reward and about how every choice we make is, on some level, an assessment of risk and reward. In fact, if you were betting on a coin toss, the risk (what you would lose if you are wrong) and the reward (what you would win if you are right) would be the only two things that should be considered. Since there is a 50% probability that the coin will land on heads and a 50% probability that it will land on tails, the probability of winning is equal to the probability of losing. Therefore, if someone tells you that they will bet you $20 against your $10 on a fair coin toss, that bet should be a no-brainer for anyone who has $10 to spare. The chances of winning are 50/50, and the reward is twice as much as the risk. Hopefully, a bet like this should set off some common sense bells in your head.

Unfortunately, most decisions we make on a daily basis do not have 50/50 odds of a favorable outcome. I mentioned in a previous discussion that, on any given roulette spin, the expected value of a $1 bet is negative 5.3 cents. Where does that number come from? The calculation of expected value comes from a fairly simple formula:

E(x) = x1p1 + x2p2 + … + xkpk

Stay with me now- it’s not as bad as it looks. First off, “E(x)” is the expected value (the solution to the problem). “x1,” “x2,” and any other x’s represent all the possible outcomes of the choice. In the coin toss example, there would only be x1 (win $20) and x2 (lose $10). “p1,” “p2,” and any other p’s represent the probability (in decimal form) of that outcome. Again, in the coin toss example, the chances are 50/50, so both p1 and p2 would be 0.50. So for the coin toss example, our equation would look like this:

expected value = (20)*(0.50) + (-10)*(0.50)

Notice how the $10 is negative because that would be money you would lose if you lost the coin toss. Some quick arithmetic produces an expected value of $5. So what does this value mean? This value meas that this is the average value that you can expect to achieve each time you make this particular coin toss bet. Of course, expected value is not telling you anything about what will happen if you take the bet only one time. If you take the bet one time, $5 will have nothing to do with the outcome- you will either win $20 or lose $10. What expected value is telling you is that, if you take the coin toss bet a million times, your wins will average out to around $5 per bet. Therefore, using the expected value equation, if you made this coin toss bet one million times, you would expect to win $5 million.

Now let’s return to roulette and work through why casino gambling is a losing proposition in the long-run. I will keep this discussion simple and ignore the fact that there are countless types of bets that you can make in roulette and that most casinos have a $5 or $10 minimum bet. In this scenario, we will be betting $1 for one spin on our lucky number: 14. So let’s start to fill in our expected value formula for this single roulette spin. First off, what happens if we lose? We lose $1. Big deal, right? So x1 equals negative 1. Now we need to determine p1 (the probability of losing). Since there are 36 black and red numbers, as well as green 0 and 00 on a typical American roulette wheel, the chances of the ball dropping on 14 on any single spin are 1 out of 38. The chances of it dropping on something other than 14 are 37 out of 38. Dividing 37 by 38 gives a p1 of 0.9737. In other words, there is a 97.37% chance that we will lose.

Now let’s consider what happens if we win. The typical payout for a single number win is 35 to one, meaning out $1 bet would win $35 if the ball drops on 14. So x2 is 35. Since the chances of winning are 1 out of 38, p2 would be 1/38 or 0.0263 (2.63%). Therefore, our expected value equation should look like this:

E(x) = (-1)*(0.9737) + (35)*(0.0263)

Solving the equation yields an expected value of -$0.0532 per bet. Therefore, for every dollar we bet on every spin of the wheel, we can expect our average loss to be 5.3 cents. Of course, there’s no way of predicting how the ball will fall on any given spin on any given night. However, if you make 200 $10-dollar roulette bets in a night, your total expected value for the evening would be a $106.40 loss.

The fun of playing roulette is the chance involved in each spin. Expected value is simply a prediction tool and a best approximation of the outcome. Expected value doesn’t account for luck. No equation can predict that 14 will hit at least 4 times in the first 50 spins when my wife and I go to Vegas in a couple of months. Any and all money I win while I’m there will not be a result of mathematical calculation, but will rather strictly be a result of my lucky Irish blood!