In the previous chapters, we’ve analyzed the probabilities of poker, bridge, and keno with combinatorial math. By looking at even simple card genealogies mathematically, we’ve begun to see things we’ve been missing. In Chapter 5, we move on to another key concept: binomial distribution and the law of averages. These concepts are key tools in the mathematical analysis of casino games, and they allow us to mathematically verify whether systematic bets like martingales and cancelations really work, strategies that many of us have believed in at one time or another, and that we may still believe in today. The power of math in gambling is about to be realized.
Repeated Games and Mathematical Expectations: Odds in Probability
Can you calculate the odds of winning?
Casino games are essentially games of chance. For example, what happens if you calculate the probability of getting a ‘0’ 3 times out of a total of 37 numbers from 0 to 36 in European roulette? The probability of getting a 0 on a single spin is 1/37. The probability of getting it 3 times in 10 repetitions is calculated via a binomial distribution as follows
- p(3 zeros in 10 spins) = C(10,3) × (1/37)^3 × (36/37)^7 ≈ 0.00192
In other words, you’ll get a ‘0’ 3 out of 10 times with a 1/500 chance.
This works for a variety of games, including the probability of winning exactly $6 at craps, the probability of flipping a coin eight times and getting heads four times, and the probability of flipping a die 12 times and getting a 5 or 6 at least four times. The formulas are different, but the idea is the same: calculate the “probability of success a certain number of times out of a given number of times” – that’s the binomial distribution.
Binomial distribution, and the “law of averages”
The binomial distribution works very well for games with independent trials (n) with a fixed probability (p), such as in a casino. For example, in a game of roulette in Las Vegas where you bet on Red (p ≈ 0.474), the binomial distribution produces the following probability curve
- Bet 4: The most symmetrical form
- Bets 8, 16, and 32: Getting closer to a bell shape
- Bet 64: Can be approximated by a normal distribution
Mathematically, these distributions are centered by their mean (np) and variance (npq), and as the number of times increases, they converge to a bell-shaped distribution by the central limit theorem. This is where the so-called “law of averages” comes into play.
However, there’s a pitfall to this rule.
“Eventually converging to the mean” means that the outcome is reliably predicted, not that you will win.
The math behind the casino: What do you need to calculate if you want to win?
The book doesn’t just throw math formulas at you; rather, it asks you practical questions to help you understand
- Spin the roulette wheel 64 times and bet $1.
→ Profit if you win 33 times or more. What is the probability of that?
→ Use Z-score → Z = (33 – np) / √(npq) - What is the probability of earning exactly $20 out of 64 times?
→ Calculated as the difference between the z-values in a given interval. - What is the probability of winning $40 or more in 500 games?
→ Calculate the Z-score to get above expected return
The conclusion is clear. As the number of times increases, it becomes increasingly difficult to expect to make a profit in a game with a house edge.
System Betting: Is there really a ‘winning formula’?
Martingale strategy
“If you lose, you double your bet, if you win, you start all over again”.
- For example, if you bet $1 on the first hand and lose, double it to $2 on the next hand, and so on to $4, $8, $16, $32, $64, and so on.
- The key to this method is that if you win even once, you can make up for all your losses and leave with a $1 profit.
- Example: If you lose a total of 6 times in a row, you lose $1 + $2 + $4 + $8 + $16 + $32 = $63. If you bet $64 on the 7th time and win, you receive $64 and recover $63, for a $1 profit.
- Although it only has about a 2.1% chance of failing,
- The problem is that there are betting limits, and so is your bankroll – in fact, it only takes a few losses in a row before you can’t keep betting anymore.
Cancellation strategy
“You make a list and adjust your bets based on that number, in order to hit a target profit you set.”
- For example, if you set a target amount of 4, 7, 1, 3, 4, 2, you would use 6, the sum of the leading and trailing numbers, as your first bet.
- If you lose the game, add the amount you just bet to the end of the list and start betting forward + backward again.
- Conversely, if you win, it removes the leading and trailing numbers you just used to bet from the list.
This strategy is structured so that when you win, you get closer to your target amount, and when you lose, your list gets longer and you wager more money. > While this may seem favorable in the short term, it can become very risky if you accumulate a losing streak, as your bets get bigger and your losses bigger.
Conclusion: Fixed Odds Games Are ‘Mathematically Unwinnable’
The math is clear in the books. No matter what system you use, no matter how you divide your bets, you can’t change your expectations as long as the fixed probabilities hold.
In other words, unless it’s a “game of chance” like blackjack or poker, where the odds change depending on the situation, your strategy is an illusion.
The Exception: What strategies really work in blackjack?
Blackjack is different
Blackjack is not binomially distributed; it’s a non-fixed probability game where the odds change as cards are dealt. Because of this, the following strategies work.
- Card counting
- Insurance bets based on the dealer’s open card
- Changing your strategy after seeing your deck composition
The book references famous mathematician Edward O. Thorp’s book, Beat the Dealer, and shows how real-world math-based blackjack strategies have made money.
FAQ: A short Q&A for those curious about casino math
No. It means ‘the outcome stabilizes,’ not ‘the player wins.’
If funds were infinite and there were no betting limits, it would be theoretically possible. In reality, it is impossible.
No. Games with a house edge accumulate losses the longer you play them. Converging to the average means approaching a ‘mathematically predetermined loss’—neither winning nor losing.
In blackjack, the composition of remaining cards affects probability because used cards are removed from the deck. In contrast, counting is meaningless in roulette or dice games because the odds are identical every round.
Finalize
Chapter 5 of The Mathematics of Games and Gambling explores the possibilities and limitations of many casino strategies through the concept of the “binomial distribution” and states bluntly
“Unless you change your expectations, you can’t win in the long run.”
But it also offers hope.
The only strategies that really win are those that respond to the “moment the odds change”.
In the next post, we’ll move on to Chapter 6, Game Theory, and enter the world of strategic thinking: poker, bluffing, and the prisoner’s dilemma.
Stay tuned for one of the keys to casino math: games of psychology and strategy.
