1. Gambling for Profit Aaron Rogers

2. How can gambling be profitable? Simple… by making +EV bets. That is, bets with a Positive Expected Value. This is how casinos make money. Casinos are nothing more than professional gambling entities. A person can do the same thing!

3. +EV bets? Say you have a bet with a friend that goes like this: a coin is flipped, and if it lands on heads, you pay your friend $10, but if it lands on tails, your friend pays you $20. (0.5 x -$10) + (0.5 x +$20) = +$5 In the long run, for every time this bet is made, you make $5! This is also called your expectation.

4. How do casinos do this? (Almost) all casino games have an inbuilt edge for the house. E.g. betting $25 on black in roulette. Winning spin = +$25, losing spin = -$25. However, the green spaces on the wheel create unequal odds of winning. Thus: (18/38 x $25) + (20/38 x -$25) = -$1.32 Negative expected value!

6. Is it really that simple? Unfortunately, not always. Sometimes, it is nearly impossible to determine a person’s expected winrate. If this is the case, we may have to estimate based on sample size. Be careful, though. A too-small sample can give misleading interpretations…

8. But can’t I still go broke, even if only making +EV bets? YES! Just because you expect you expect to make $5 from each coin toss in the long run, doesn’t mean that you will survive the short run. This is why bankroll management is important. Casinos, obviously, have gigantic “bankrolls” relative to the bets they make with their customers.

9. What is an appropriate bankroll? This depends on your win rate (expectation), the standard deviation of that win rate, and your personally defined “Risk of Ruin.” “Risk of Ruin,” or RoR, is the chance that, if a person is playing a +EV game with a chance to lose, he/she will go broke before an infinite number of trials.

10. So how do we calculate it? First we need to know what the win rate and standard deviation are. Let’s take the same coin flip example from earlier. Recall that our expectation is $5/flip. This can be also called our average, or µ:

11. Finally, the calculation: Required bankroll = -(σ²/2µ)ln(RoR) σ² = Variance = Standard deviation², which is $225 or $15² µ = Expection/expected value/average/mean, which is $5 Ln is the natural logarithm function RoR is Risk of Ruin. For our purposes, let’s say it’s 5% Plugging in our numbers yields: -($15²/(2*$5))ln(0.05) = $67.40

13. So, now that I have a +EV game and a bankroll that I’m comfortable with, what next? Go and make money! However, be sure to avoid the pitfalls of gambling professionally: Swings will be experienced, and they can be wild. Try not to let it affect your mental state. Always remember to stick to the math… avoid –EV spots unless they will lead to larger +EV spots.

14. Extra questions… Can I increase my EV by predicting the next outcome, based on previous outcomes? Answer: NO! This is called the Gambler’s Fallacy. The reason it doesn’t work is because each trial is INDEPENDENT from each other. The law of large numbers looks at future trials, not past trials. In the coin flip example, if you start out by getting heads 5 times in a row, the chance of a tail result in the next flip is STILL 50%. What if I take a –EV bet, but structure my betting so that every time I lose, I bet more. Thus, once I win, I win back everything I lost? Answer: This is called the Martingale system. It works… except it relies on the assumption the person using it has an infinite bankroll. Unfortunately, NOBODY has an infinite bankroll.

15. Screw the math, I just want to play! That’s fine! Just realize that you’re playing for fun, not for profit. In the long run, you *will* lose money if you are making –EV bets. As long as the entertainment makes up for it, then all’s well! Even if you ignore the math, at least keep to this general rule: NEVER gamble what you can’t afford to lose! If you want to play for profit, then realize that math is behind EVERYTHING. Learn it, and use it to your advantage!