“Probability? Isn’t that just luck?”
In Chapter 2 of The Mathematics of Games and Gambling, the author breaks down the concepts of probability and expected value using real-world gambling problems to make them accessible to the math-challenged. If you’re dreading reading this because it reminds you of school math class, don’t worry. I’ll cut to the chase.
Why Probability Matters 1: Your Gut Feelings Can Be Wrong
Probability is so important in gambling because it translates into a result called “odds” that you can’t know through mere luck and intuition.
The 17th-century French nobleman de Mere often made these bets.
- Roll 1 die 4 times and win if you get a 6 at least once.
The odds were in his favor, but he changed it and started losing money.
- Roll 2 dice 24 times and win if you get a double 6 at least once.
It looks similar, but in reality, the latter was less than a 50% chance, which goes to show how you can lose money like this if you don’t know the math and just trust your gut.
Why probability matters 2: If the game stops, how will the winnings be divided?
The other reason probability is important, besides win rate, is when the game is interrupted. Jimmy and Walter are playing a 5-point game and are at 4:3 when suddenly the cops show up and the game is stopped. If they can’t play the rest of the game, how would it be fair to divide the money?
If you do the math, Jimmy has a 3/4 chance of winning, so it makes sense for him to keep 75% of the winnings and Walter 25%. In gambling, probability-based distribution is more compelling than morals or emotions. Without the absolute math of probability, we’d have a mess of people fighting over who gets more money.
One thing to remember instead of complicated calculations: expectations
So, if you had to pick just one concept of probability that you need to know when playing casino games, what would it be? The most important one is “expected value”.
What is Expectation?
- The idea is that when you have multiple outcomes, you average them out to see “how much you make or lose per hand”.
For example, let’s say you flip two coins and get the following results
- 2 dollars for two fronts
- 3 for two backs
- 4 for each
You don’t need to do all the math on this, you just need to know that the result is that you lose an average of $0.75 per hand.
The bottom line is that if your expectation is negative, the more you do, the more you lose.
Las Vegas Roulette, Why Do Casinos Love It?
A roulette table has 38 spaces, including 0 and 00, so if you bet money on a number, your chance of winning is 1/38. But the casino only pays you on a 1/35 basis if you win. This difference is the house edge, or the casino’s cut.
- In reality, they take 2.6% of your wager every hand.
- The game is different, but almost all bets are designed to be this unfavorable.
3 key formulas you don’t need to know
- Probability of something not happening = 1 – Probability of something happening
- Probability of A or B happening = Probability of A + Probability of B (but not at the same time)
- Probability of A followed by B = Probability of A x Probability of B (but only when they are independent events)
Just remember: if they don’t happen at the same time, add; if they happen in order, multiply.
An easy example to understand expectations
In the book, there’s a fun example: suppose you offer a discount, say $15 if you pre-register and $20 if you pay at the door, but there’s no refund if you don’t show up. It’s definitely cheaper if you buy now, but there’s a chance you won’t be able to show up, so you’ll lose money. The calculation is that it pays to pre-register if the probability is greater than 75%. Another example is Pascal’s famous bet on the existence of God. For example, even if the probability of God’s existence is low, the expected value is infinite if the believer will receive an eternal reward. This leads to the philosophical logic that we should always choose the side with the greater gain, even if the probability is low.
FAQ: Things you don’t need to know about math
Yes. Avoid games with negative expected value, and simply understanding why the odds are low can be helpful.
That’s correct. Roulette, slot machines, and even blackjack are all designed mathematically to always favor the house.
Theoretically, it’s a fair game, but in reality, fees or mistakes can ultimately result in a loss.
No. It’s sufficient to simply understand whether this game has a positive or negative expected value, and how many times you need to play to turn a profit.
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In the next installment, we’ll move on to real-world games. Dice-based games like backgammon, craps, and Chuck-a-Luck are the kind of games you’d find in a real casino or at home. Let’s take a look at how the rules and math work together and some simple strategies.
