The Mathematics of Gambling:. Probability, Psychology, and the Thrill…

Jul 07, 2023

Roberto Lentini

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Casino game creators expertly exploit humanity’s fascination with low-probability events and tantalize ‘what if’ fantasies. The thrill of the unknown has often enchanted humanity. Despite this allure, casinos hold the house’s edge, prompting any curious mathematicians to wonder why the casino always wins and where we can unearth the secrets to tipping the odds, if even possible, in our favor. From the every day to the extraordinary, probability unveils the enigmatic nature of luck itself. Probability governs our lives; whether it’s the probability of accidentally sharing clothing choices with a friend at a party or the chance of slipping on a banana peel while strolling down the street, it can all be mathematically explained.

The attraction of low-probability events captivates our imagination; they become the center of our wildest fantasies, greatest fears, and most unimaginable excitements. These rare and consequential happenings combine the realms of mathematics and psychology. This interdisciplinary problem becomes quite complex as human behavior seems to outshine mere logic. The blend of human interest, fear, and excitement that arises from these low-chance events has even paved the way for entire industries like insurance and gambling, which both model their businesses on giving dollar value to chance.

Through an exploration of the captivating nature of gambling, the intricacies of probability, and the profound influence of human psychology in games of chance, this article aims to uncover the secrets behind the house’s edge. I will shed light on why the casino seemingly always wins, and ultimately reveal strategies that could tip the odds in our favor, even in the face of statistical complexity and game creators’ opacity.

Let’s take a closer look at the human obsession with low odds in the context of gambling. To better understand games of chance and basic probability, let’s play a simple gambling game that involves flipping a coin. In this game, I’ll flip a fair coin and if it lands on heads, I’ll double your money. However, if it lands on tails, I’ll keep the money you gave me.

You might wonder how much money am I expected to win playing this game? Well, it turns out that over an infinite number of plays you’ll find yourself with the same amount of money you started out with [1].

The expected winnings of this game are the probability of winning times how much you win minus the probability of losing times how much you lose [1]. In this case, if your starting bet is 10$, it’s (0.5 * 10) — (0.5 * 10) = 0. You walk away with no winnings and no losses. The expectation of the game ensures that over the long run, your wins and losses balance out and you maintain the same amount you started with.

Let’s imagine a variation of the classic American roulette wheel where we strip away the numbers and focus solely on 18 red spaces and 18 black spaces. Like the coin flipping game, betting on the correct color grants a doubling of the wager. Initially, this setup offers a seemingly fair 50% chance of doubling our money. If we bet $100, the expected winnings would be: (0.5 * $100) — (0.5 * $100) = $0. Just like the coin flip example, in the long run, we’d expect to walk away with the same amount we started with.

However, casinos aren’t in the business of offering odds that don’t favor them. To ensure profitability, the roulette wheel introduces two additional green spaces strategically tipping the odds in the casino’s favor. This is also known as the house’s edge. With these green spaces in play, the odds of winning when betting on red or black shift to 47.36%. Consequently, if we bet $100 on black over a large number of plays, our expected outcome would be: (0.4737 * $100) — (0.5263 * $100) = -$5.26. This means we’d effectively be trading a $100 bill for $94.74.

While an individual player may experience occasional wins or losses in the grand scheme of things, the casino will consistently come out on top. This slight shift in odds may seem insignificant, but it makes a significant enough impact on the casino’s overall profits to keep them in business.

Calculating expected winnings reveals how casino games are designed to favor the house. This casino’s advantage or “the house’s edge” is the factor which leads some to Gambler’s Ruin, where losses may exhaust one’s funds completely.

Another critical concept that often eludes avid gamblers is that of independent events. At the roulette table, the belief that past outcomes influence future ones leads to the gambler’s fallacy. Whether black has hit ten times in a row or hasn’t hit in numerous spins, the odds of it landing on the next spin remain fixed at 47.37% due to the independence of each event. The confusion may arise from human emotions overpowering logic and a misunderstanding of the difference between the odds of multiple consecutive black outcomes and the isolated odds of the next spin. It’s crucial to grasp that the board has no memory, and past occurrences hold no sway over future probabilities, ensuring that the odds remain unchanged at 47.35%.

Let’s put a pin on the odds of it all and look at the psychology of gambling. Consider two options: option one guarantees you $10 with a 100% chance, while option two presents an 80% chance of receiving $12.5. Although these options may seem distinct, the underlying similarity lies in the expected winnings. The expected value for option two, calculated as 0.8 * $12.5–0.2 * $0, also equals $10. While the potential to gain an extra $2.5 might seem enticing, the key differentiating factor here is the perceived chances of losing money, highlighting the psychological aspect of decision-making in gambling.

Let’s examine another comparable scenario. In the first choice, you are offered a guaranteed $10, while the second choice entails a mere 0.1% chance of winning $10,000. Remarkably, the expected winnings for both options remain equal at $10. Yet, despite the statistical equivalence, the psychology at play becomes intriguing. The $10 prize may not carry significant impact compared to the exhilaration of the $10,000 opportunity. Risk tolerance and the thrill of a potentially colossal win enter the equation.

Let’s take a look at the lottery system Lotto Max. The way this system works is you pick 7 numbers on a ticket, the more numbers you get correct, the more money you make. Here is the price breakdown of a recent draw:

The expected winnings are calculated as such:

Expected Value = (Probability of winning Jackpot * Amount of Jackpot) + (Probability of winning 6 out of 7 numbers + Bonus * Amount of 6/7 + Bonus) + (Probability of winning 6 out of 7 numbers * Amount of 6/7) + (Probability of winning 5 out of 7 numbers + Bonus * Amount of 5/7 + Bonus) + (Probability of winning 5 out of 7 numbers * Amount of 5/7) + (Probability of winning 4 out of 7 numbers + Bonus * Amount of 4/7 + Bonus) + (Probability of winning 4 out of 7 numbers * Amount of 4/7) + (Probability of winning 3 out of 7 numbers + Bonus * Amount of 3/7 + Bonus) + (Probability of winning 3 out of 7 numbers * Amount of 3/7) + (Probability of winning a Free Play * Amount of Free Play)

Plugging in the values:

Expected Value ≈ (1/33,294,800 * $70,000,000) + (1/4,756,400 * $317,035.90)+ (1/113,248 * $4,603.00) + (1/37,749 * $912.70) + (1/1,841 * $105.80) + (1/1,105 * $105.80) + (1/82.9 * $20.00) + (1/82.9 * $20.00) + (1/8.5 * $5.00)

This gives us an expected value of approximately $3.46 per ticket, which is sold at $5 marked up at approximately 31%. This markup makes our alternative game of either $10 with a 100% chance or $10,000 with a 0.1% chance seem truly remarkable. The odds at the roulette table also appear more favorable, with an average loss of only $5.26 on a $100 gamble, in contrast to the $17.78 loss on a $100 worth of lottery tickets. It’s fascinating to observe that individuals may opt for a less favorable decision with lower odds, enticed by the promise of a significantly larger sum despite its diminished value.

This phenomenon delves into the realm of psychology, moving beyond mere statistics. Several factors could contribute to this behavior, such as the perception of reduced risk when spending less on a lottery ticket compared to joining the roulette table. Additionally, the allure of high rewards, such as fantasizing about winning $70 million, may overshadow the prospect of doubling a modest $5 through a higher-chance bet.

The problems we have tackled so far have been relatively straightforward due to their lack of “distraction variables” or noise. When we look back at the examples of the coin flip, roulette, and lottery each event is independent from each other, winning/losing once won’t guarantee another win/loss. Furthermore, there are minimal factors that affect each spin, so each spin we can assume has the same probability of experiencing a certain event.

However, the prior assumptions we can make in games of chance become much more complex when we delve into the world of sports betting. The difficulties that come when calculating expected winnings for sports betting is due to its dependence on human performance and there are many factors that can influence this. For instance, a player’s past performance may offer insights into future performances, but external elements like injuries or personal issues can affect their games. Sports games are also not independent from each other: consecutive losses might lead to a losing mentality which can potentially increase the odds of another loss. All of these factors need to be considered for each player on any team, which makes it a very challenging task to calculate probabilities.

In the context of roulette, the house gains its edge from the additional green spaces marked 0 and 00 which tip the odds in favor of the casino. However, in sports betting, where uncertainties and dependencies abound, the betting companies implement various strategies to maintain a balanced betting market and secure profitability.

For instance, in a hockey game where the last-place team faces the first-place team, the betting company faces the challenge of creating balance. They designate the first-place team as the strong favorite, leading to less potential winnings for the gamer. On the other hand, they set higher potential reward on the last-place team to attract potential bets on the underdog. The betting company relies on bettors’ behavior, hoping lots of individuals will place bets on the underdog expecting a substantial payout in case of an upset, known as the longshot bias. But even if a bettor consistently places bets on the favored team, a single loss could require many more wins by betting on the favored team to recover the lost money due to the small incremental gains made in such scenarios.

The best odds you have of winning in gambling would be to play skill based games that are combined with luck, and sport betting is one of the best options. The challenges that make it hard for an individual to reduce noise in sports betting are the same the betting company has, hence why they take fees and modify payout to keep the house’s edge. But there are very good ways of making money sports betting through a hedging strategy.

Hedging is a risk management strategy that prioritizes reducing potential losses while sacrificing some potential profits. While this approach may yield smaller gains in a single bet, it provides more consistent and secure outcomes over time. The essence of hedging lies in placing a bet that is opposite to your initial wager.

Let’s consider hedging a future bet. Imagine at the start of the season you placed a bet on Team A to win the Super Bowl. You place a $100 bet with odds of +500 (5-to-1), which means you stand to win $500 if Team A is victorious. Now the playoffs final have come around and your team has a chance of winning. To safeguard your investment, you can place a hedge bet of $100 on Team B with odds of +300. If Team B wins, you’ll receive $300, offsetting the initial $100 loss on Team A. On the other hand, if Team A wins you lose the $100 hedge bet but still make a $400 profit from the original wager.

In in-play hedging, the strategy involves adjusting bets during a game to adapt to changing circumstances. A notable example is the “middle” betting strategy in hockey. Suppose you initially bet on the Toronto Maple Leafs (+2.5) in their away game against the Montreal Canadiens which means you are expecting the Leafs to win outright or lose by two goals or less. Between the second and the third period, with the Maple Leafs holding a 3–1 lead and Montreal being favored by 1.5 goals for the third period you can execute the middle strategy. Placing a new bet on Montreal (-1.5) during the second intermission creates a middle zone where both bets can win. If the Leafs hold their lead or win, your original bet on Toronto (+2.5) succeeds, while the second bet on Montreal (-1.5) loses. If the Maple Leafs lose by one goal, your original bet loses, but the second bet wins, resulting in a win for both bets. However, if the Leafs lose by two goals, both bets lose. This strategy serves as a hedge in case the Leafs’ lead slips away, offering a chance to win with the Montreal bet, while simultaneously maximizing your profit if the initial prediction holds true.

The temptation of uncertain outcomes captivates our imagination, enticing us with dreams of big wins. However, the house’s edge and betting companies’ tactics ensure they maintain an advantage. Games of chance and probability have values tied to them which make some much better than others while still assuring the house generally wins. While hedging in sports betting can reduce risk and is probably one of the best ways to earn money gambling, it’s essential to remember that, in the end, the odds are designed to favor the house.

Thank you for reading the article and supporting my work. If you want to get my future articles right in your mailbox make sure you follow my content and subscribe to my mailing list.

[1] Q: If you double your bet every time you lose, won’t you eventually win and come out ahead? Ask a Mathematician / Ask a Physicist. (2018, April 7). https://www.askamathematician.com/2018/04/q-if-you-double-your-bet-every-time-you-lose-wont-you-eventually-win-and-come-out-ahead/

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