Life Lessons from Poker is a seven-part series of articles about how the things we learn in our efforts to master poker can turn out to be indispensable life tools away from the felt. If you’re coming to the series for the first time, Part 1 can be found here.
Let’s say I flip a coin ten times, and it comes up heads every time. What are the odds that it comes up heads again on the next flip?
You’re probably rolling your eyes at the question, because you’ve almost certainly heard this thought experiment in probability dozens of times already and know the answer is that it’s still 50%. But what if I tell you that it’s not?
I deliberately left out one key word in my phrasing of the question, namely “fair,” as in “a fair coin,” meaning one which flips heads and tails with equal probability. Of course, when you phrase the question that way, it’s tautological: The setup asserts that the coin always flips with 50/50 probability, and the question then asks what the probability is on a flip.
The reason for this thought experiment, as we all know, is to disabuse people of the notion that a 50/50 chance guarantees a 50/50 distribution of outcomes, that a coin doesn’t “owe” you tails just because it’s produced more heads than expected in the past.
In reality, there are no perfectly fair coins. If a coin flips a certain way over a very large sample, you may come to believe that it’s deliberately rigged. Even ignoring this possibility, however, normal coins have different embossed designs on the two faces, which means that the metal isn’t perfectly evenly distributed. And even if a coin is designed to be totally fair, there are manufacturing defects to account for, and the possibility that the flipper’s technique may tend to produce an odd or an even number of half-flips ever so slightly more often.
Assuming that you have confidence that the coin was not deliberately designed to be unfair, you shouldn’t set the odds of another heads flip very far from 50%. Nonetheless, the more heads that you flip in a row, the higher the chances that it does in fact weight towards heads at least slightly. If you’re betting at even money on repeated flips of the same coin, then, it is in fact rational to bet on whichever result has come up more often in the past.
Between superstition and reason
Looked at that way, the coin flip thought experiment is a much stronger metaphor for life. The world isn’t made up of “fair coins” with their probabilities clearly marked on them. It’s full of complex things and complex people, all of whose behavior depends on what is often an uncountably large number of variables. We cut through that tangle by making simplifying assumptions about how things work – that a coin flips heads and tails with equal probability, for instance – and, hopefully, update those assumptions in the face of new evidence.
Unfortunately, when it comes to choosing those assumptions, and to updating them, many people tend to go wrong in one of two ways.
Most of the population, especially those without a lot of formal education, tend to rely on their instincts, but as I’ve discussed in a previous column, our instincts tend to make us superstitious. Someone relying purely on instinct to attempt to predict outcomes will tend to make assumptions too quickly and put too much faith in them based on a small number of observations. They’ll then fall prey to confirmation bias, which hinders their ability to update their assumptions based on new information.
Superstitious thinking isn’t the only way to go wrong, however. There are also those who’ve been taught to think critically, but take the concept of scientific provability to an extreme. They demand that things be proven conclusively to be true before accepting them as so, and place little value on their own experiences and those of others until the weight of evidence is enough to meet their standards of provability.
“Statistical significance” belongs in scientific papers, and “reasonable doubt” is a concept best left in the courtroom. Outside of exacting fields like these, demanding near-100% certainty before accepting something as true means not putting valuable life experience to use, and is the sort of thing that makes one “book smart but street stupid.”
Fortunately, there is a middle ground between instinct and rationality, which is intuition. A well-developed intuition provides a reality check for our instincts, while reminding the rational mind that those instincts exist for a reason. Finding the sweet spot can be tricky, but there’s a handy mathematical tool to help us out.
Bayes’ Theorem is a pretty simple equation with deep implications. It tells you the extent to which you should update an assigned probability based on new information. It sounds really confusing when laid out in the abstract, so let’s look at a concrete example first, then give the general formulation.
Poker players lend each other money all the time. It’s an essential part of the job and the culture, because dealing with swings is difficult otherwise, and no one’s going to lend you money if you aren’t willing to do likewise. However, scammers and degenerates abound, and there’s always a chance that money lent is not going to be returned.
In many cases, the scamming party doesn’t just make off with the first loan they get; it’s often after several loans have already been taken and repaid. There can be warning signs that a disappearing act is coming up, however; a classic one is slow payment. However, honest people can also end up being slow to pay due to circumstances beyond their control, so, it’s reasonable to wonder how likely it is that someone is a scammer, based on them being slow to repay the first loan we give them.
Bayes’ Theorem tells us that we can calculate this probability if we have reasonable guesses at the following other probabilities:
- The odds that any given poker player is a scammer. (That is, the percentage of the community that has or will one day scam someone).
- The frequency with which slow payment occurs in general.
- The frequency with which a scammer is slow to pay.
Notice that although we need three probabilities to calculate one, which seems like a step backwards, these are all general probabilities that we can guess at based on experience with the poker world and the stories we’ve heard. Conversely, if this is our first financial dealing with the person in question, we have no direct information about their odds of being a scammer. That’s why Bayes’ Theorem is useful.
Let’s say we assume 10% of poker players have scammed or will scam, that slow payment happens with 15% of all loans within the poker community, but that a scammer will tend to be slow to pay 45% of the time they borrow money.
We therefore assign a prior probability of 10% to this person being a scammer, and want to know by how much we adjust this now that we’ve experienced them being slow to pay. The logic is really very simple: at 45% to 15%, we’ve estimated that a scammer is three times more likely to be slow to pay than the overall average (including both scammers and non-scammers). Therefore, we multiply our prior probability by three, and should regard it as now 30% likely that the person in question intends to scam us in future.
More generally, we can write Bayes’ Theorem this way:
P(A|B) = P(A) x P (B|A) / P(B)
Where P(A) and P(B) are the overall probabilities of A and B being true independently, while P(A|B) means how often A is true in cases where we know B is, and P(B|A) means how often B is true in cases where we know A is.
I told you that it was a little confusing in the abstract, but look at it in the context of the example and it should be fairly clear.
Bayes and the bookies
Although this series is about life lessons from poker, I feel I should digress briefly here to point out that Bayes’ Theorem is hugely important in the world of sports betting. If you want to try to predict how well a given batter is going to hit against a given pitcher, it’s naïve to look only at their past history. Rather, their direct matchup batting record should be adjusted based on their own individual records, plus more general probabilities about how, say, left-handed batters stack up against right-handed pitchers overall.
Both the bookies and the professionals who try to beat them employ this kind of thinking very explicitly, and these days, usually with computer assistance.
Bayesian intuition in poker
Of course, almost no one actually carries around a notebook and calculator to put Bayes’ Theorem into practice explicitly at the poker table, or in day-to-day life. But those who’ve developed a good intuition are applying Bayesian logic unconsciously all the time. The first time this acquaintance approached us, we weighed the risk of getting scammed against the social benefits of lending the money and deemed it worth it. If they’re coming back to us for a second loan after being slow to pay back the first one, we revise that risk estimate up in a Bayesian manner and re-evaluate.
Naturally, this sort of thinking is hugely important in developing reads at the poker table, especially within the first few orbits of sitting down. Perhaps in your first hand at the table, the player in middle position three-bets against an under-the-gun open, but the hand doesn’t get to showdown. You certainly can’t say for sure that this is an aggressive player, as even the nittiest of nits gets dealt Aces once in a while, and many would argue that it’s far too soon to have any sort of read. But if you’re familiar with the field, you probably have an estimate of how many aggressive players you’re likely to run into, and how much more inclined those guys are to three-bet against early position opens; if you’re making good use of your intuition, you’re already revising upwards your assessed probability that the guy is going to be aggressive in future.
Of course, such intuition is all the more useful when you’re combining information from several sources, none of which is solid evidence on its own. Say the player in question is a young man with expensive headphones around his neck, who’s talking shop with his buddy in the next seat and has several large stacks of low-denomination chips in front of him. Combined with the three-bet, those observations would make you a lot more confident in expecting aggression from him. If it were a middle-aged white guy with a smaller collection of chips, telling a bad beat story to no one in particular, this might be contradictory evidence and result in you discounting the three-bet for the time being until you’ve seen him play some more hands.
Bayesian intuition in life
There’s a funny parallel between the first orbit reads in poker and what is probably one of the most common situations in which adults apply Bayesian intuition in life, even if they don’t realize it. What I’m talking about are first dates.
It’s not the most romantic notion, that both you and the person sitting across from you in the coffee shop are both mentally crunching probabilities to infer things about each other based on a comparison of minor observations to lived experience. But that’s pretty much exactly what you’re doing, albeit at an unconscious or semi-conscious level.
Small talk isn’t all that small on a first date, after all. You’re not asking about the other person’s job, family and pets just to fill what would otherwise be an awkward silence; you’re sketching an outline of their personality using the answers, and filling in the details using your Bayesian intuition. If she says she’s an accountant, you’ll infer something different about her personality outside of work than if she says she’s a dancer, or a social worker. If he’s got a cat, maybe you have beliefs about how a cat-owner differs from a dog-owner, or someone who keeps reptiles, or doesn’t have any pets at all. You’re also looking at physical details, most likely, especially in terms of clothing and personal style, since people often signal things about themselves this way, deliberately or not.
Of course, this is a skill just like developing reads at the poker table, but people with significant romantic experience become so good at it that concepts like speed dating and Tinder are predicated on the idea that experienced daters can get most of what they need to know from a dozen miscellaneous facts and a few photos or a brief face-to-face encounter.
The same sort of approach can serve you well in other contexts as well: Evaluating office environments while job-hunting, for instance, or getting a feel for a town or neighborhood while contemplating a move. Depending on your own temperament and outlook, you may prefer to approach the situation as if you were placing a sports bet, reading a poker table, or evaluating a first date. In essence, however, all of these amount to the same thing, which is using your intuition to strike the correct balance between your gut instincts and your rational appraisal of the situation.
Alex Weldon (@benefactumgames) is a freelance writer, game designer and semipro poker player from Montreal, Quebec, Canada.