The Gambler’s Ruin: The Mathematics of Losing

The gambler’s ruin is a statistical fact that despite any strategy or betting system, you will eventually lose your money gambling. This is because of the simple fact that the mathematical odds of winning a bet, and the payout itself don’t match up. Casinos implement house edge in all of their bets, regardless of which casino game or sports bet you pick.

That is not to say you can’t make money off the casino, but you must understand, the games are not designed to favour you over the house. It is crucial to understand the relation between probability and expected value. By learning how the casino calculates its odds, you can come up with a betting system to avoid the inevitable and make the most of any luck that comes your way. The stars will not always align, but you need to be prepared for when they do, and figure out how you can weather the variance.

Basic Model of the Gambler’s Ruin

The Gambler’s Ruin uses the random walk model, which defines how your bankroll changes over time. We can calculate these outcomes using the Monte Carlo method, which simulates hundreds of thousands (if not more) of outcomes. The idea is that the statistical anomalies and variance all level out after a huge volume of rounds. And then at those points, the results should line up with the expected value of the wagers. That means you will lose money in proportion to the RTP on your wager, and eventually lose all your money.

Defining Expected Value

The expected value is how much money you will make if your bet wins. But it will always be a little larger than the real possibility of winning. This is because the house applies a little juice, or vig, skimming a little money off your winnings. It is easiest explained with roulette bets. Let’s take the smallest and biggest bets, namely, the 1:1 wagers (red/black, high/low, odds/evens), and the straight up wagers (bet on a single number).

Playing a game of European roulette, you have 37 segments on the wheel. When playing a 1:1 wager, you cover 18 of those segments. This means, your real chances of winning are 48.64%, whereas the expected value from the odds is 50%. That additional 1.35% is the house edge. If you play 37 rounds of roulette and the ball lands on each of the 37 segments (a mathematically perfect sequence), and spend $1 per spin, you will be at a loss. You spend $37 in total for all the spins, and only 18 win – meaning you will win a total of $36.

The straight up bets are priced at 35:1, but you are only covering 1 from the 37 segments. That means, your real chances of winning are 2.70%, but the implied probability is 2.77%. Should you win 1 round from 37 tries, you will spend $37 and win $36 again.

Defining House Edge

The house edge is made by providing you with negative expected value bets. It doesn’t matter if you are playing banker bets in zero commission baccarat, adding safety cash outs in your favourite crash game, or playing a 99% RTP slot machine. In card games or roulette, we can definitively calculate this edge. Simply by counting the exact probability of winning and measuring it against the expected value’s implied probability. But in slot machines and crash games, it is not as apparent.

These games have RTP values, provided by game auditors who thoroughly test each game’s algorithms. They are designed to provide completely randomized results, but a house edge remains. We can only theoretically calculate the chances of each bet winning based on RTP and the payout amount for each individual winning outcome.

Losses in Theory vs Practise

While a statistical eventuality, the gambler’s ruin doesn’t consign you to losing every time you gamble. It may take millions of spins on a slot machine, or cards drawn at a blackjack table for the outcomes to properly reflect the actual probabilities. This is because some statistical anomalies and frequencies cannot be precalculated.

The key word here is variance. This is the amount and frequency of the winnings that deviates from what is mathematically optimal. If you were to flip a coin 10 times, and it landed on heads and tails in alternating flips, there is no variance. These are the statistically sound outcomes, and at the end you should have 5 heads and tails outcomes each. However, if the coin landed on heads 3 times in a row, and then landed on tails for the following 7 times, there is suddenly a great variance.

These are the anomalies that can occur by chance. Winning or losing streaks are created by variance, and there is no way of knowing when the streak will end, or a new one will start. In the long run, the streaks and variance should diminish, as the results balance out. But that may take an infinite number of rounds to achieve. In the short run, variance is a lot higher, and anything can happen.

Dangers of Variance

Though variance can play tricks on gamers. We can be lured into thinking that a table is “hot”, or that we can predict the next outcome based on what happened in the previous rounds. These are all gambler’s fallacies, and make similar kinds of assumptions as the gambler’s ruin. But the latter is based on maths and fact, whereas the fallacies are misconceptions. And the casino wants to encourage your fallacies – it is a way to get you to spend more, enhancing your chances of losing more money to the house.

Gambler’s Ruin and Variance in Sports Betting

The gambler’s ruin can be applied to virtually every form of gambling. It is not simply something that you will experience in casino games. The Gambler’s Ruin is not as straightforward in sports betting as it is in casino games. The casino has no say in what happens on the field in a sports game. They define the odds based on previous game statistics, historical data, trends and other metrics. Oddsmakers feed the data into complex algorithms to generate the odds. But some so many external factors and variables can come into play.

For instance, we don’t know what’s going on inside the coach’s mind when they pick their lineups, or how confident the players are. Variance is generally a lot higher in sports betting than casino gaming. Especially when the odds on a bet are overestimated, or severely underestimated. Expert sports bettors don’t just pick out bets based on who they think will win. Analysing the odds and finding golden opportunities is just as part of the game as picking a moneyline winner. At the end of the day, you are relying on pure chance – something casino gamers are all too familiar with.

The sportsbooks always add a little juice to your winnings. It ensures that in the long run, the house will make its money. Plus, it makes it more difficult for hedge bettors to bet on all outcomes and guarantee a profit.

Key Takeaways to Avoid Gambler’s Ruin

The gambler’s ruin theory does factor in variance, but it is not definite. It takes the assumption that after countless hands, the results will reflect the actual probabilities. In the long run, these numbers will come close to the theoretical ones, but it may take millions of rounds to get there.

The gambler’s ruin can be offset by adding a few tricks of your own. Such as changing your stake after each round. By using progressive betting systems, you can minimize your losses when the variance is high. Alternatively, some systems such as the martingale system can aggressively push to make a profit. But these are dangerous, as they can lead to tremendous losses if you can’t keep up.

Another way of avoiding the gambler’s ruin is by implementing targets. Specifically, targets that are quite achievable, and don’t require huge variance. You can therefore quit while you are ahead, or accept a set amount of losses, and avoid further losses.

How to Plan Your Gambling Bankroll

Variance can be your greatest ally, but you need to recognise when it is working in your favour. When luck swings your way and you get a string of wins, you will get the perfect opportunity to quit on a high. But if you are losing, you have to decide whether it is worth weathering the storm or quitting with whatever money you have left.

In either case, a bigger bankroll is needed to maintain your gaming. Ideally, you should have enough money to last a long gaming session without going bust. You won’t be able to find bets with positive expected value at casino games unless you are counting cards in blackjack, but even then you need a higher percentage of 10s left in the deck to gain a slight advantage. Therefore, you need to rely on variance. And considering how it can also play against you, be prepared for losses along the way. But if you hit your targets and leave early, you may just be able to get into the green.