Now the expert player can compare those odds by imagining this hand one hundred times. The expert doesn’t know if she is going to win or lose this hand, but she does know that if she played this exact same hand one hundred times, she would, on average, win twenty times, collecting $100 with each victory, yielding $2,000.
And she knows that playing one hundred times will cost her only an additional $1,000 (because she has to bet only $10 each time). So even if she lost eighty times and won only twenty times, she would still pocket an extra $1,000 (which is the winnings of $2,000 less the $1,000 needed to play).
Got it? It’s okay if you don’t, because the point here is that probabilistic thinking tells the expert how to proceed: She is aware there’s a lot she can’t predict. But if she played this same hand one hundred times, she would probably end up $1,000 richer. So the expert makes the bet and stays in the game. She knows, from a probabilistic standpoint, it will pay off over time. It doesn’t matter that this hand is uncertain. What matters is committing to odds that pay off in the long run.
“Most players are obsessed with finding the certainty on the table, and it colors their choices,” Annie’s brother told her. “Being a great player means embracing uncertainty. As long as you’re okay with uncertainty, you can make the odds work for you.”
Annie’s brother, Howard, is competing in this Tournament of Champions right alongside her when the FossilMan is eliminated. Over the past two decades, Howard has established himself as one of the finest players in the world. He has two World Series of Poker bracelets and millions in winnings. Early in the tournament, Annie and Howard lucked out and didn’t have to directly compete for many big pots. Now, however, seven hours have passed.
First the FossilMan was eliminated by that bit of bad luck. Another competitor named Doyle Brunson, a seventy-one-year-old nine-time champion, was knocked out by a risky attempt to double his chips. Phil Ivey, who won his first World Series of Poker tournament at twenty-four, was eliminated by Annie when she drew an ace and queen against Ivey’s ace and eight. Over time, the players at the table have dwindled until there are only three players remaining: Annie, Howard, and a man named Phil Hellmuth. It is inevitable Annie and Howard will butt up against each other. The contestants spar over chips and hands for ninety minutes. Then Annie gets a pair of sixes.
She starts tallying what she does and doesn’t know. She knows she has strong cards. She knows, from a probabilistic standpoint, that if she played this hand one hundred times, she would do okay. “Sometimes when I’m teaching poker, I’ll tell people there are situations where you shouldn’t even look at your cards before you bet,” Annie told me. “Because if the pot odds are in your favor, you should always make the bet. Just commit to it.”
Howard, her brother, seems to like his hand as well, because he pushes all of his chips, $310,000, onto the table. Phil Hellmuth folds. The bet is to Annie.
“I’ll call,” she says.
They both turn over their cards. Annie reveals her pair of sixes.
Howard reveals a pair of sevens.
“Nice hand, Bub,” Annie says. Howard has an 82 percent chance of winning this hand, collecting chips worth more than half a million dollars, and becoming the table’s dominant leader. From a probabilistic perspective, they both played this hand exactly right. “Annie made the right choice,” Howard later said. “She committed to the odds.”
The dealer turns over the first three communal cards.
“Oh, God,” Annie says and covers her face. “Oh, God.”
The six and the two queens in the communal pile give Annie a full house. If Annie and Howard replayed this hand one hundred times, Howard would likely win eighty-two of those contests. But not this time. The dealer puts the remaining cards on the table.
Howard is out.
Annie jumps from her chair and hugs her brother. “I’m sorry, Howard,” she whispers. Then she runs out of the studio. She starts sobbing before she makes it to the door.
“It’s okay,” Howard says when he finds her in the hall. “Just beat Phil now.”
“You have to learn to live with it,” Howard told me later. “I just went through this same thing with my son. He was applying to colleges and he was nervous about it, so we came up with a list of twelve schools—four safety schools, four he had an even chance of getting into, and four that were stretches—and we sat down and started calculating the odds.”
By looking at the statistics those schools had published online, Howard and his son calculated the likelihood of getting into each college. Then they added all those probabilities together. It was fairly basic math, the kind even English majors can manage with a little bit of Googling. They figured out that Howard’s son had a 99.5 percent chance of getting into at least one school, and a better than even chance of getting into a good school. But it was far from certain he would get into one of the stretch schools, the ones he had fallen in love with. “That was disappointing, but by going through the numbers, he felt less anxious,” Howard said. “It prepared him for the possibility that he wouldn’t get into his first choice, but he would definitely get in somewhere.
“Probabilities are the closest thing to fortune-telling,” Howard said. “But you have to be strong enough to live with what they tell you might occur.”
III.
In the late 1990s, a professor of cognitive science at the Massachusetts Institute of Technology named Joshua Tenenbaum began a large-scale examination of the casual ways that people make everyday predictions. There are dozens of questions each of us confront on a daily basis that can be answered only with some amount of forecasting. When we estimate how long a meeting will last, for instance, or envision two driving routes and guess at which one will have less traffic, or predict whether our families will have more fun at the beach or at Disneyland, we’re making forecasts that assign likelihoods to various outcomes. We may not realize it, but we’re thinking probabilistically. How, Tenenbaum wondered, do our brains do that?
Tenenbaum’s specialty was computational cognition—in particular, the similarities in how computers and humans process information. A computer is an inherently deterministic machine. It can predict if your family will prefer the beach or Disneyland only if you give it a specific formula for comparing the merits of beach fun versus amusement parks. Humans, on the other hand, can make such decisions even if we’ve never visited the seaside or Magic Kingdom before. Our brains can infer from past experiences that, because the kids always complain when we go camping and love watching cartoons, everyone will probably have more fun with Mickey and Goofy.
“How do our minds get so much from so little?” Tenenbaum wrote in a paper published in the journal Science in 2011. “Any parent knows, and scientists have confirmed, that typical 2-year-olds can learn how to use a new word such as ‘horse’ or ‘hairbrush’ from seeing just a few examples.” To a two-year-old, horses and hairbrushes have a great deal in common. The words sound similar. In pictures, they both have long bodies with a series of straight lines—in one case legs, in the other bristles—extruding outward. They come in a range of colors. And yet, though a child might have seen only one picture of a horse and used only one hairbrush, she can quickly learn the difference between those words.
A computer, on the other hand, needs explicit instructions to learn when to use “horse” versus “hairbrush.” It needs software that specifies that four legs increases the odds of horsiness, while one hundred bristles increases the probability of a hairbrush. A child can make such calculations before she can form sentences. “Viewed as a computation on sensory input data, this is a remarkable feat,” Tenenbaum wrote. “How does a child grasp the boundaries of these subsets from seeing just one or a few examples of each?”
In other words, why are we so good at forecasting certain kinds of things—and thus, making decisions—when we have so little exposure to all the possible odds?
In an attempt to answer this question, Tenenbaum and a colleague, Thomas Griffiths, devi
sed an experiment. They scoured the Internet for data on different kinds of predictable events, such as how much money a movie will make at the box office, or how long the average person lives, or how long a cake needs to bake. They were interested in these events because if you were to graph multiple examples of each one, a distinct pattern would emerge. Box office totals, for instance, typically conform to a basic rule: There are a few blockbusters each year that make a huge amount of money, and lots of other films that never break even.
Within mathematics, this is known as a “power law distribution,” and when the revenues of all the movies released in a given year are graphed together, it looks like this:
Graphing other kinds of events results in different patterns. Take life spans. A person’s odds of dying in a specific year spike slightly at birth—because some infants perish soon after they arrive—but if a baby survives its first few years, it is likely to live decades longer. Then, starting at about age forty, our odds of dying start accelerating. By fifty, the likelihood of death jumps each year until it peaks at about eighty-two.
Life spans adhere to a normal, or Gaussian, distribution curve. That pattern looks like this:
Most people intuitively understand that they need to apply different kinds of reasoning to predicting different kinds of events. We know that box office totals and life spans require different types of estimations, even if we don’t know anything about medical statistics or entertainment industry trends. Tenenbaum and Griffiths were curious to find out how people intuitively learn to make such estimations. So they found events with distinct patterns, from box office grosses to life spans, as well as the average length of poems, the careers of congressmen (which adhere to an Erlang distribution), and the length of time a cake needs to bake (which has no strong pattern).
Then they asked hundreds of students to predict the future based on one piece of data:
You read about a movie that has made $60 million to date. How much will it make in total?
You meet someone who is thirty-nine years old. How long will he or she live?
A cake has been baking for fourteen minutes. How much longer does it need to stay in the oven?
You meet a U.S. congressman who has served for fifteen years. How long will he serve in total?
The students weren’t given any additional information. They weren’t told anything about power law distributions or Erlang curves. Rather, they were simply asked to make a prediction based on one piece of data and no guidance about what kinds of probabilities to apply.
Despite those handicaps, the students’ predictions were startlingly accurate. They knew that a movie that’s earned $60 million is a blockbuster, and is likely to take in another $30 million in ticket sales. They intuited that if you meet someone in their thirties, they’ll probably live another fifty years. They guessed that if you meet a congressman who has been in power for fifteen years, he’ll probably serve another seven or so, because incumbency brings advantages, but even powerful lawmakers can be undone by political trends.
If asked, few of the participants were able to describe the mental logic they used to make their forecasts. They just gave answers that felt right. On average, their predictions were often within 10 percent of what the data said was the correct answer. In fact, when Tenenbaum and Griffiths graphed all of the students’ predictions for each question, the resulting distribution curves almost perfectly matched the real patterns the professors had found in the data they had collected online.
Just as important, each student intuitively understood that different kinds of predictions required different kinds of reasoning. They understood, without necessarily knowing why, that life spans fit into a normal distribution curve whereas box office grosses tend to conform to a power law.
Some researchers call this ability to intuit patterns “Bayesian cognition” or “Bayesian psychology,” because for a computer to make those kinds of predictions, it must use a variation of Bayes’ rule, a mathematical formula that generally requires running thousands of models simultaneously and comparing millions of results.*2 At the core of Bayes’ rule is a principle: Even if we have very little data, we can still forecast the future by making assumptions and then skewing them based on what we observe about the world. For instance, suppose your brother said he’s meeting a friend for dinner. You might forecast there’s a 60 percent chance he’s going to meet a man, since most of your brother’s friends are male. Now, suppose your brother mentioned his dinner companion was a friend from work. You might want to change your forecast, since you know that most of his coworkers are female. Bayes’ rule can calculate the precise odds that your brother’s dinner date is female or male based on just one or two pieces of data and your assumptions. As more information comes in—his companion’s name is Pat, he or she loves adventure movies and fashion magazines—Bayes’ rule will refine the probabilities even more.
Humans can make these kinds of calculations without having to think about them very hard, and we tend to be surprisingly accurate. Most of us have never studied actuarial tables of life spans, but we know, based on experience, that it is relatively uncommon for toddlers to die and more typical for ninety-year-olds to pass away. Most of us don’t pay attention to box office statistics. But we are aware that there are a few movies each year that everyone sees, and a bunch of films that disappear from the theaters within a week or two. So we make assumptions about life spans and box office revenues based on our experiences, and our instincts become increasingly nuanced the more funerals or movies we attend. Humans are astoundingly good Bayesian predictors, even if we’re unaware of it.
Sometimes, however, we make mistakes. For instance, when Tenenbaum and Griffiths asked their students to predict how long an Egyptian pharaoh would reign if he has already ruled for eleven years, a majority of them assumed that pharaohs are similar to other kinds of royalty, such as European kings. Most people know, from reading history books and watching television, that some royalty die early in life. But, in general, if a king or queen survives to middle age, they usually stay on the throne until their hair is gray. So it seemed logical, to Tenenbaum’s participants, that pharaohs would be similar. They offered a range of guesses with an average of about twenty-three additional years in power:
That would be a great guess for a British king. But it’s a bad guess for an Egyptian pharaoh, because four thousand years ago people had much shorter life spans. Most pharaohs were considered elderly if they made it to thirty-five. So the correct answer is that a pharaoh with eleven years on the throne is expected to reign only another twelve years and then die of disease or some other common cause of death in ancient Egypt:
The students got the reasoning right. They intuited correctly that calculating a pharaoh’s reign follows an Erlang distribution. But their assumption—what Bayesians call the “prior” or “base rate”—was off. And because they had a bad assumption about how long ancient Egyptians lived, their subsequent predictions were skewed, as well.
“It’s incredible that we’re so good at making predictions with such little information and then adjusting them as we absorb data from life,” Tenenbaum told me. “But it only works if you start with the right assumptions.”
So how do we get the right assumptions? By making sure we are exposed to a full spectrum of experiences. Our assumptions are based on what we’ve encountered in life, but our experiences often draw on biased samples. In particular, we are much more likely to pay attention to or remember successes and forget about failures. Many of us learn about the business world, for instance, by reading newspapers and magazines. We most frequently go to busy restaurants and see the most popular movies. The problem is that such experiences disproportionately expose us to success. Newspapers and magazines tend to devote more coverage to start-ups that were acquired for $1 billion, and less to the hundreds of similar companies that went bankrupt. We hardly notice the empty restaurants we pass on the way to our favorite, crowded pizza place. We become trained, in other words, t
o notice success and then, as a result, we predict successful outcomes too often because we’re relying on experiences and assumptions that are biased toward all the successes we’ve seen—rather than the failures we’ve overlooked.
Many successful people, in contrast, spend an enormous amount of time seeking out information on failures. They read inside the newspaper’s business pages for articles on companies that have gone broke. They schedule lunches with colleagues who haven’t gotten promoted, and then ask them what went wrong. They request criticisms alongside praise at annual reviews. They scrutinize their credit card statements to figure out why, precisely, they haven’t saved as much as they hoped. They pick over their daily missteps when they get home, rather than allowing themselves to forget all the small errors. They ask themselves why a particular call didn’t go as well as they had hoped, or if they could have spoken more succinctly at a meeting. We all have a natural proclivity to be optimistic, to ignore our mistakes and forget others’ tiny errors. But making good predictions relies on realistic assumptions, and those are based on our experiences. If we pay attention only to good news, we’re handicapping ourselves.
“The best entrepreneurs are acutely conscious of the risks that come from only talking to people who have succeeded,” said Don Moore, the Berkeley professor who participated in the GJP and who also studies the psychology of entrepreneurship. “They are obsessed with spending time around people who complain about their failures, the kinds of people the rest of us usually try to avoid.”
This, ultimately, is one of the most important secrets to learning how to make better decisions. Making good choices relies on forecasting the future. Accurate forecasting requires exposing ourselves to as many successes and disappointments as possible. We need to sit in crowded and empty theaters to know how movies will perform; we need to spend time around both babies and old people to accurately gauge life spans; and we need to talk to thriving and failing colleagues to develop good business instincts.
Smarter Faster Better: The Secrets of Being Productive in Life and Business Page 19