American Scientist

A mathematician, a philosopher, and a gambler walk into a bar. As the bartender pulls each of them a beer, he decides to stir up a bit of trouble. He takes a die from his pocket and rolls it ostentatiously on the bar counter. It comes up one.

The mathematician says: “The probability that one would come up is one in six, and on the next throw the probability will be the same. If we roll the die infinitely many times, the relative frequency of the number one will converge to one in six—that is, to one occurrence every six throws.”

The philosopher strokes her chin and remarks: “Well, this doesn’t mean we won’t get the same number with the next throw. Actually, it’s physically possible to have the same number on the next 1,000 throws, although that’s highly improbable.”

The gambler says: “I know you’re both right, but I wouldn’t bet on one for the next throw.”

“Why not?” asks the mathematician.

“Because I trust mathematics, and so I expect one to come up about once every six throws,” the gambler answers. “Having the same number twice in a row is a rare event. Why would that happen right now?”

The gambler’s argument is a mix of conceptual inadequacy, misinterpretation, irrelevant application of mathematics, and misleading use of language. She thinks that she has information that will increase her chances of winning—that there are now five numbers to choose from instead of six, and as such the randomness of the game is “losing its strength.” This sort of belief reinforces a gambler’s impulse to bet—it won’t make her quit the game, but rather will encourage her to continue gambling.

Some people believe that confronting problem gamblers with mathematics—a kind of mathematical counseling, often called “facing the odds”—can help them overcome their addiction. After all, since our earliest school days, many of us have learned to trust mathematics as the provider of necessary and logical truths. But we also trust our senses, as well as the patterns we discern from our experiences and the words we use to communicate with one another. Mathematics has its own language, and the extent to which we should trust mathematics depends on how we interpret these words, especially when applied to physical reality. Indeed, understanding gamblers’ relationship to math reveals something deeper about the nature of mathematics itself.

Why the House Always Wins

I don’t gamble, and I suspect few mathematicians do—or at least, they don’t gamble to get rich. In my youth, I was fascinated by games of chance and loved to play them so I could watch probability at work. But after studying mathematics and its philosophy in more depth, my interest in such games vanished. I came to see them as simply mathematical models wearing sparkling clothes. Instead, I wanted to figure out how I could help other people to see gambling in the same way.

All casino games—whether based on pure chance such as roulette, craps, and slots, or skilled card games such as poker or bridge—rely on certain basic statistical and probabilistic models. Uncertainty is built into them, which is what makes the games fun to play and explains their continued existence. But casino games would never run if the house wasn’t confident that they’d always win in the end. The mathematics of the games, including their rules and payout schedules, ensures that the house will profit in aggregate, regardless of individual behavior.

In mathematical terms, this guarantee is expressed through the fact that the house edge (the casino’s share of the overall revenue in a game coming from players’ bets) is positive. The expected value of a bet is defined as follows:

(probability of winning)×(payoff if you win)−(probability of losing)×(amount lost if you lose)

The house edge of a game is defined as the opposite of the expected value calculated for all possible bets (house edge = −expected value). For example, in European roulette, a wheel spins and you have to guess into which numbered compartment a small ball will land. There are 37 numbered compartments (0 to 36). If you bet $1 on one number (called a straight-up bet), the payoff is 35 times what you bet, and the probability of winning is 1 in 37. So the expected value of that bet is:

(1/37)×$35−(36/37)×($1)

That is about −$0.027 or, as a percentage, 2.7 percent of the initial bet as a loss. Expected value can be read as an average; in our example, you might expect that, over the long run, you would lose on average $2.70 every 100 games. The house edge in European roulette—the house’s share of all the income produced by that game in the form of bets over the long run—is 2.7 percent.

From a player’s point of view, a positive house edge should mean that she can’t make a living off that game; over the long run, the house will have an advantage. That’s why a pragmatic principle of safe gambling behavior is, When you make a satisfactory win, take the money and get out of there.

The gambler’s argument in the opening story illustrates a spectrum of misconceptions that fuels games of chance. There’s the so-called gambler’s fallacy, where someone believes that a series of bad plays will be followed by a winning outcome, in order for the randomness to be “restored.” Then there’s the conjunction fallacy, when the gambler estimates the probability of a combination of events to be higher than the probability of one of those events—such as when someone uses addition (rather than multiplication) to estimate the probability of two or more independent events. For example, in sports betting, someone might bet once on several “almost sure” outcomes occurring together, thinking that it’s likely that all the house’s favorite teams will win—ignoring the fact that the product of the probabilities of several wins is a number significantly lower than the probability of any individual win.

Another gambling misconception is the near-miss effect, when an outcome differs just a little from a winning one, which induces the gambler to believe that she was “so close” that she should try again. Here you might think of slots, scratch cards, or the lottery, where such events are the most frequent, but virtually any game of chance produces them. The near-miss effect involves incorrectly estimating probabilities but is also linked to other conceptual inadequacies regarding conditional probabilities and time dependence. In such cases, the gambler mentally splits the winning outcome into matching and nonmatching parts—an action that’s mathematically irrelevant—and develops an overconfidence in a new occurrence of the matching part in a future play. This fallacy ignores the probabilities of the occurrence of the nonmatching part and of the conjunction of the two predicted events; the “so close” is actually “so far” in probability terms and figures.

Naturally, the gambling industry makes the most of such fallacies. And all such cognitive distortions are recognized as important risk factors for problem gambling.

The Neuroscience of Gambling

Gambling has existed since antiquity, but in the past 30 years it has grown at a spectacular rate, turbocharged by the internet and globalization. Problem gambling has grown accordingly, and has become particularly prevalent in the teenage population. Even more troublingly, a study in 2013 reported that more than 90 percent of problem gamblers don’t seek professional help. Gambling addiction is part of a suite of damaging and unhealthy behaviors in which people participate despite warnings, such as smoking, drinking, or compulsive video gaming. It draws on a multitude of cognitive, social, and psychobiological factors.

Psychological and medical studies have found that some people are more likely to develop a gambling disorder than others, depending on their social condition, age, education, and experiences such as trauma, domestic violence, and drug abuse. Problem gambling also involves complex brain chemistry. Gambling stimulates the release of multiple neurotransmitters including serotonin and dopamine, which in turn create feelings of pleasure and the attendant urge to maintain them. Serotonin is known as the happiness hormone and typically follows a sense of release from stress or fear. Dopamine is associated with intense pleasure, released when we’re engaged in activities that deserve a reward, and precisely when that reward occurs—such as seeing the roulette ball land on the winning number, or hearing the sound of the slot machine showing a winning line.

For the most part, gambling addiction is viewed as a medical and psychological problem, but this knowledge hasn’t resulted in widely effective prevention and treatment programs. The lack of successful interventions might be because research has often focused on the origins and prevalence of addiction, rather than on the cognitive premises and mechanisms that take place in the brain.

Some problem gambling programs frame the distortions associated with gambling as an effect of poor mathematical knowledge. Some clinicians argue that reducing gambling to mere mathematical models and bare numbers—without sparkling instances of success and the adventurous atmosphere of a casino—can lead to a loss of interest in the games, a strategy known as reduction or deconstruction. They promote warning messages along the lines of, “Be aware! There is a big problem with those irrational beliefs. Don’t think like that!” But whether this kind of messaging works is an open question. Beginning a couple of decades ago, several studies were conducted to test the hypothesis that teaching basic statistics and applied probability theory to problem gamblers would change their behavior. Overall, these studies have yielded contradictory, inconclusive results, and some found that mathematical education yielded no change in behavior. So what’s missing?

Combating Cognitive Distortions

One problem is that interventions to promote mathematical literacy among gamblers generally push the message that gamblers should unconditionally trust mathematics. But the pitfall for many gamblers isn’t so much a lack of trust in mathematics as much as an incorrect application and interpretation.

The limitations of mathematical counseling make sense when we recall that the mathematics of real-world events are far from pure numbers; rather, they take the form of descriptions, strategies, predictions, and expectations, all mediated by language and meaning. By making a distinction between pure and applied mathematics, between truths that are necessary and those that are contingent, and by noticing how often we mix mathematical and nonmathematical terms, we might steer ourselves on the right track to correct our cognitive distortions.

No single expert or guide can help us here. We need the combined wisdom of the mathematician, the philosopher, and the psychological counselor to help combat the forces that sustain problem gambling. Indeed, some of the associated cognitive distortions tap into genuine philosophical debates, such as the meaning of randomness, which is uncontrolled and proceeds without any rules. Mathematicians and philosophers struggle to agree on a rigorous and universally accepted definition of randomness, despite the centrality of the concept to probability theory.

In the early 20th century, mathematicians Émile Borel and Richard von Mises attempted to define a random sequence, such as what would occur if you flipped a coin repeatedly under precisely the same conditions, and wrote down the results as a sequence of zeros and ones (zeros for the heads, and ones for the tails). The resulting data is known as a trivial random sequence. Borel and von Mises tried to describe in mathematical axioms the fact that there’s no rule for such a sequence, and each new term is independent of what went before—but other mathematicians and philosophers objected to their definitions. Some critics argued that the empirical setup was indispensable to the description but couldn’t itself be a mathematical object; other objections were more fundamental.

The philosophical complexity of applied mathematics makes it clear that simply learning to trust mathematics is a naive prescription for gambling addiction. Applied mathematics involves establishing an equivalence between the empirical, real-world context (target domain) and abstract mathematical structures (source domain). To do so, we need to idealize away from the messiness of the real world with models. After “doing the math” with the help of the mathematical theory in the source domain, the derived mathematical truths are interpreted back in the target domain via our models, where we can create predictions, representations, and descriptions.

When we put abstract, formal mathematics in empirical situations such as games of chance, we ultimately rely on language to express newly inferred relations as truths. However, these “truths” are no longer necessary truths; they depend on meanings, interpretations, and context. As such, they are contingent truths. If we’re too zealous in abstracting or idealizing our empirical context, or if we poorly interpret the mathematical truths in the target domain, such modeling can lead to erroneous results. In these situations, it’s not pure mathematics that’s to blame, but the whole setup.

Any application of mathematics is a balance between relevance and convenience—a choice, a refinement, and finally a cross-checking against the real world. A successful real-world application relies as much on a mathematician’s or scientist’s intuition as it does on scientific or mathematical rigor. We should not blindly trust applied mathematics—which is not to say that we don’t trust the pure mathematics behind it, but simply that we need to be mindful of the place of language and interpretation.

The same goes for the mathematics of gambling. When describing games or making predictions about bets, we use mathematical models, and any such model depends on language for interpretation and empirical validation. Take the statement, “The relative frequency of the die showing a one will converge toward one in six with the number of throws.” One interpretation is the gambler’s expression that, “I expect a one to come up about once every six throws.” Here, about means on average or approximately—but this wording does not reflect the complete mathematical meaning of converge, which assumes an infinite series of experiments for the limit to be approached. Moreover, it is the relative frequency (the ratio between the number of a variable’s occurrences and the number of experiments) that approaches that limit and not the absolute frequency (the total number of a variable’s occurrences), which seems to be the reference of the gambler’s words. We might say in this case that we have a poor model, due to interpretation and language, and it would be empirically invalidated when a one comes up twice in a row in a certain period of time.

Perhaps most problematically for gamblers, statistical models are grounded in probability theory, one of the fields in mathematics most open to philosophical debate. Probability carries various senses and meanings. You might think of it in terms of the recorded relative frequency of the occurrence of an event, such as if a roulette ball landed on a red number in 38 percent of the plays during a period of time. Or you might take it to be the payout percentage that the game offers (usually called game odds, such as 3:2 in betting on the outcome of a match, which would translate to a 40 percent probability of losing). Or you might perceive it as a physical feature of the game in question, its inner potential to produce a certain frequency of outcome. For example, if you rolled a pair of dice over and over, and noted that the pair five and two appeared 150 times in 3,000 plays, you might think of it as a property of the game, which then gives a game-dependent probability of 5 percent.

However, probability’s standard mathematical meaning is so-called Kolmogorovian probability, a way of measuring likelihood with the concepts of measure theory. There are, however, other concepts of probability, such as inductive, propensic, subjective, frequentist, or classical (Laplacian). All these versions are also mathematical in nature—and perhaps surprisingly, many gamblers’ subjective perceptions of the concept of probability more or less match some of these theoretical concepts. Other statistical terms can be imported and used in ordinary language too, such as expectation, mean, or average, with their usual, not purely mathematical meanings.

Perhaps the biggest difficulty with using mathematical concepts in the context of gambling is that all probability theory is grounded in the idea of infinity—yet all of our gaming experiences are finite. This inconsistency lies behind many cognitive distortions, including those of the gambler in the opening anecdote. She evaluated the probability of one coming up twice in a row on the basis of (finite) past observations, extending what was rare for her to being rare in general. The problem is amplified by words that carry a different meaning in mathematical and nonmathematical contexts. An illustrative example is the concept of an event: In mathematics, it’s a formal element of a set, having nothing to do with the complexity of what an event means in real life.

Overall, if we think that mathematics can provide any sort of fix for problem gamblers, then we must be careful. It’s certainly an important cognitive asset for gamblers to know just how unlikely winning outcomes are—some with probabilities close to zero, which would mean they’d have to play for several lifetimes, sometimes in the order of thousands, to get close to a probability of one. Nonetheless, “facing the odds” or learning to trust mathematics often isn’t enough. The gamblers also need to trust the role of representation and description in the mathematical models, while being careful when interpreting the real-life predictions obtained through these theoretical models. That balance requires a sophisticated grasp of which models and idealizations are adequate for the reality they try to describe, and which are irrelevant or misleading. Because gambling concerns the relationship between mathematics and reality, which also relies upon language, the issue falls within the philosophy of mathematics and of mathematical modeling.

Getting to the bottom of our gambler’s misunderstanding of probability is likely to require a conversation between the mathematician and the philosopher, who in turn need to advise the cognitive psychologist about how to talk to her client. Such an interdisciplinary venture isn’t necessarily as difficult a task as it seems: After all, cognitive psychology and the philosophy of knowledge and language share a common intellectual boundary in many respects.

Sometimes truth is not as straightforward as being validated or invalidated by empirical evidence or even scientific facts. Sometimes it originates in the very nature of the arguments we make, including the language we use to express them. Mathematics has its own language, and the truths of applied mathematics are sensitive to the way we understand and express them. The cognitive distortions associated with gambling are a relevant example of such sensitive truths. What’s remarkable is that fighting these distortions reveals something about both the nature of mathematics and the nature of human understanding—and that knowing when not to trust mathematics is as crucial as knowing when to trust it.

This article was adapted from an essay previously published in Aeon, aeon.co.