The Math In Broke: Mathematical Expectation

In Broke: The New American Dream, you’ll see traders, poker players, investors, and politicians. Each has a unique manner in earning money, yet each relies on the concept known as mathematical expectation.

If you want to trade, learn this important concept first. Knowing it will explain the argument of Accuracy versus Expectation.

Accuracy Model: a high percentage of winning trades. Very hard to do for any period of time.

Expectation Model: a lower percentage of winning trades, 40% for example, but winners that are several multiples the size of the losers. Easier to accomplish in practice, although harder to deal with emotionally b/c our school systems have ingrained in us that 90% is an “A” Student, and 40% is a failure. Batting .400 though is Hall of Fame-level performance.

Mathematical expectation is used by everyone from the folks state lotteries, in all forms of advertising, Las Vegas and Atlantic City, insurance underwriters, and Wall Street traders. Without it, we would not know if our models would be profitable. It is an important tool in helping us put our statistical analysis to work.

Here is a brief lesson in how mathematical expectation works.

On the roulette wheel there are 36 numbers, double zero, and the blank. That makes 38 spaces to bet on. Each bet costs $1 to play. The winner pays $35. To calculate the mathematical expectation of the roulette wheel you do the following:

Multiply the probability of winning by what you win when you win. And from that, you subtract the probability of losing by the cost of each bet. The difference is the mathematical expectation. If it’s positive, it’s a fair bet. If it’s negative, you don’t play.

[(1/38) x (35)] – [(37/38) x (1)] = mathematical expectation of playing roulette.

(35/38) – (37/38) = (-2/38) or (-1/19).

So in the case of the Roulette wheel, the best bet is not to play. The problem is playing Roulette is fun! Most professional money handlers don’t find losing money fun. And that’s the difference between the professional and an amateur.

The roulette wheel, keno, and state lotteries are examples of games with negative expectation. Another famous one is the slot machine. If you go to Vegas with a budget constraint — and fixed amount that you’re willing to lose — and you play a game of negative expectation, you are guaranteed to go home broke. You can develop a slight edge in poker and black jack.

Professional traders develop models to buy and sell securities or commodities. They create entry rules, exit rules, and position sizing rules. They run 10 to 20 years worth of data through the model and come up with hypothetical results. Of course they use professional simulation software. The models can take several minutes to run.

The hypothetical results include the number of winning trades, the number of losing trades, percent winning trades, the percent losing trades, the biggest loss, the biggest win — just to name a few. With this data, traders can calculate the mathematical expectation of a trade and determine whether or not the system is worth following.


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