Math Problems
How's your bracket? Mine, too.
Tom Stoppard’s Rosencrantz and Guildenstern are Dead presents Shakespeare’s Hamlet from the bewildered point of view of two of the Bard’s bit players, the comically indistinguishable nobodies who become headliners in Stoppard’s brilliant play. The show opens before our heroes have even joined the action in Shakespeare’s epic. They have been “sent for” and are marking time by flipping coins and getting heads each time.
Guildenstern keeps tossing coins and Rosencrantz keeps pocketing them. Significantly, Guildenstern is less concerned with his losses than in puzzling out what the defiance of the odds says about chance and fate. “A weaker man might be moved to re-examine his faith, if in nothing else at least in the law of probability.”
The coin-tossing streak depicted provides us with a chance to consider these probabilities. Guildenstern offers among other possible explanations the one mathematicians and investors should favor: “a spectacular vindication of the principle that each individual coin spun individually is as likely to come down heads as tails and therefore should cause no surprise each individual time it does.” In other words, past performance is not indicative of future results.
Even so, how unlikely is a streak of this length?
The probability that a fair coin, when flipped, will turn up heads is 50 percent (the probability of any two independent sequential events both happening is the product of the probability of both). Thus, the odds of it turning up twice in a row is 25 percent (½ x ½), the odds of it turning up three times in a row is 12.5 percent (½ x ½ x ½), and so on. Accordingly, if we flip a coin 10 times (one “set” of ten), we should only expect to have a set end up with 10 heads in a row once every 1024 sets.
Rosencrantz and Guildenstern got heads nearly 100 consecutive times. The chances of 100 in a row are 1/7.9 x 1031. In words, we could expect it to happen once in 79 million million million million million (that’s 79 with 30 zeros after it) sets. By comparison, the universe is about 13.9 billion years old, in which time only about [1 with 17 zeros after it] seconds have elapsed.
Looked at another way, if every person who ever lived (around 110 billion of us) had flipped a 100-coin set simultaneously every second since the beginning of the universe (again, about 13.9 billion years ago), we should expect all 100 coins to have come up heads two times.
If anything like that had happened to you (especially on a bet), you’d agree with Nassim Taleb that the probabilities favor a loaded coin.1 But, then again, while 100 straight heads are less probable than 99, which are less probable than 98, and so on, any exact order of tosses is as likely (actually, unlikely) as 100 heads in a row. The maths are exactly the same. We notice the unlikelihood of 100 in a row because of the pattern. More “normal” combinations look random and thus expected.
Looked at still another way, if there will be one “winner” selected from a stadium of 100,000 people, each person has a 1 in 100,000 chance of winning. But we aren’t surprised when someone wins, even though the individual winner is shocked.
The point here is that the highly improbable happens all the time. In fact, much of what happens is highly improbable. As the late football coach Mike Leach explained about college football upsets, “Everybody’s all surprised every time this stuff happens. It surprises me everybody gets surprised because it happens every year like this that there are surprises. The most surprising thing would be if there weren’t any surprises. So, therefore, in the final analysis, none of it’s really that surprising.” Convoluted, sure, but 100 percent accurate.
These sorts of math problems are the focus of this week’s TBL.
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Math Problems
As this TBL goes live, just 16 games and one day of the NCAA Tournament are in the books, yet my bracket is a mess. I’d bet yours is, too.
My perfect bracket was destroyed with the tournament’s first game when Maryland defeated West Virginia. I missed on the second game, too, when Furman, a 13-seed playing its first tourney game since 1980, upset Virginia, a four-seed, by getting an improbable steal and nailing a long three with 2.4 seconds left.
I watch an embarrassing amount of college basketball, but I didn’t get a W on my bracket until Kansas, a one-seed, beat Howard in yesterday’s fourth game, and I only picked nine of Thursday’s 16 games correctly. Raise your hand if you had 15-seed Princeton taking out Arizona, a two [liar].
This wild randomness is why March Madness is so great.2
Nobody knows for sure what the actual numbers are, but it is estimated that as many as 100 million NCAA Tournament brackets are filled out each year. But there has never been a perfect bracket – not even close – and it’s exceedingly unlikely there ever will be. It's simple probability. The odds of picking all 63 tournament games correctly are the odds of each game multiplied together. Those are insanely long odds.
Gregg Nigl, a neuropsychologist from Columbus, Ohio, has come closest so far to a perfect performance. In 2019, submitting a hastily filled in bracket – and under the influence of cold medicine – Nigl predicted the first 49 games of the tournament correctly (into the Sweet Sixteen) before seeing his streak snapped. Nigl’s bracket finally went bust on game 50 (the third game on the second weekend) – about a one-in-five-million level of success.
Nobody made it through the first round last year or the year before. One bracket in 2017 was right through an incredible 39 games. In 2014, one bracket was alive through 34 games.
That’s why my bracket is a mess. Yours, too.
Spoiler Alert: March Madness is a high variance proposition.
Top seeds are 149-1 against 16s (Wahoo-Wa), through Thursday, in the seeded era – since 1979 – and 26 of 43 national champions were one-seeds (60%), but 12-seeds beat fives 36 percent of the time and have had at least one of their number advance in 12 of the past 14 tournaments (I’m looking at you, VCU). The 11 and 10 seeds win 39 percent of the time. Five 11-seeds have made it to the Final Four: LSU in 1986, George Mason in 2006, VCU in 2011, Loyola Chicago in 2018, and UCLA in 2021. St. Peter’s made the Elite Eight last year as a 15.
The nine beats the eight more often than not, although Villanova won a national championship as an eight-seed in 1985 (the lowest seed to win it all) and UNC, like Butler in 2011 and Kentucky in 2014 – eight-seeds all – made the final game last year. UConn won the championship as a seven in 2014. North Carolina State in 1983 and Kansas in 1988 won it all as six-seeds.
Since seeding began in 1979, all four two-seeds have reached the Sweet 16 in only six of 43 years and won’t this year thanks to Princeton. A double-digit seed has advanced to the Sweet 16 in 39 of 43 years, including each of the past 14. At least one team seeded fifth or lower has advanced to the Elite Eight in 40 of 43 years. A First Four team almost always wins a second game.
It should go without saying that NCAA basketball tournament bracket math isn’t an exact science. It is often claimed that the odds of picking a perfect NCAA tournament bracket are a staggering one in 9,223,372,036,854,775,808 (that’s 9.2 quintillion – a quintillion is a billion billion).
If every person on the planet (about 7.9 billion people) started filling out a bracket per minute, it would take over 2,000 years to fill out 9.2 quintillion. However, that calculation assumes every game is a 50:50 proposition, which of course isn’t the case.
Duke math professor Jonathan Mattingly claimed the average college basketball fan has a far better chance of achieving bracket perfection than one in 9.2 quintillion. According to Mattingly, adjusting probability based on seeding, the odds of picking all 63 games correctly is more like one in 2.4 trillion. Using a different formula, DePaul mathematician Jay Bergen calculated the odds at one in 128 billion. The NCAA says it’s one in 120 billion.
Whatever the true odds, if you were really lucky, your perfect bracket lasted about halfway through Thursday’s games.
That said, the highly unlikely happens surprisingly often. In 2009, New Jersey grandmother Patricia Demauro set a craps world record over four hours and 18 minutes by rolling a pair of dice 154 times before crapping out. Her odds of doing that were one in 5.6 billion.
In Monte Carlo, on August 18, 2013, a Roulette ball landed on black 26 times in a row – the longest such streak ever recorded, with odds against of 136,823,184 to one. As the streak lengthened, gamblers lost millions betting on red, believing that the odds changed with the length of the run of blacks – a classic exhibition of the gambler’s fallacy.
In June 2002, electrician Mike McDermott won £194,501 on the UK National Lottery after correctly choosing five numbers and the bonus ball. Fast-forward four months to October and Mike was still playing. He matched the exact same five numbers and bonus ball that he had in June, having continued to play them out of habit. He picked up another £121,157. The odds of Mike winning twice with the same numbers were over five trillion to one.
Despite remarkably long odds, Richard Lustig has won the lottery seven different times, Roy Sullivan survived being struck by lightning seven times, and the roulette wheel at the Rio in Las Vegas landed on 19 seven consecutive times.
Motoyuki Mabuchi went all-in with four aces in the main event game of the 2008 World Series of Poker but lost to a royal flush held by Justin Phillips. Even worse for Mabuchi, the final diamond ace that handed Phillips victory came with the last card drawn. As Ray Romano said, “How many times you gonna see that?!”
The odds of this exact showdown occurring were about one in 165 million. That’s a bad beat.
College GameDay basketball broadcasts on ESPN offer one lucky student the opportunity to win a lot of money by making a half-court shot.3 The bottom of the net has somehow been found on such 47-foot heaves four times in a row at the University of Virginia.
It seems like it ought to be easier than that to pick winners and better brackets. There is no spread involved and a full season of games has been played, allowing us to evaluate the teams with a significant amount of data. But the short answer is that there are simply too many variables and too much randomness involved to think that we can succeed in picking all those winners.
There are more ways to arrange a 52-card deck than there are atoms on Earth, which means that every time you shuffle a deck of cards, the same order of cards has probably never been seen in human history and may never be seen again.
Note to self: The global economy has many more than 52 variables.
As Han Solo said, “Never tell me the odds.”
Daniel Kahneman describes randomness as noise and is convinced that “bias has been overestimated at the expense of noise.” Moreover, “noise and bias are independent sources of error, so that reducing either of them improves overall accuracy. . . and the procedures by which you would reduce bias and reduce noise are not the same. Indeed, “as a first rule, there is more noise than people expect, and there’s more noise than they can imagine because it’s very difficult to imagine that people have a very different opinion from yours when your opinion is right, which it is.”
“And you find variability within individuals, depending morning, afternoon, hot, cold. A lot of things influence the way that people make judgments: whether they are full, or whether they’ve had lunch or haven’t had lunch affects the judges, and things like that.”
Despite that reality, according to a study undertaken by the CFP Board, 88 percent of consumers and 30 percent of financial advisors think it’s easy to manage money. However, everyone should agree that predicting what will happen in the world and in the markets has many more variables and requires much more complex thinking than predicting the binary outcomes of basketball games.
Reminder: We aren’t very good at picking basketball games.
As Churchill said, “All this shows how much luck there is in human affairs, and how little we should worry about anything except doing our best.”
Comprehensive research has shown that even professionals are prone to being taken in by recent outcomes – hiring “hot” (lucky) money managers rather than those with more skill. A better approach – in investing and elsewhere – is (as Sam Hinkie would say) to build a good process and trust it.
As Kahneman has explained, “to compute probabilities you need to keep several possibilities in your mind at once. It’s difficult for most people. Typically, we have a single story with a theme. People have a sense of propensity, that the system is more likely to do one thing than the other, but it’s quite different from the probabilities where you have to think of two possibilities and weigh their relative chances of happening.”
We prefer to think linearly, manufacturing a storyline, in effect, with a beginning, middle, and end. That’s why we are so susceptible to the narrative fallacy. We inherently prefer stories to data. Contingencies and (perhaps random) consequences don’t correspond to the way we like to see the world. We are – pretty much all the time – either looking backward and creating a pattern to fit events and constructing a story that explains what happened along with what caused it to happen, fitting what we see or assume we see into a preconceived narrative, or both.
Dealing effectively with probabilities requires that we recognize the power of the random and contingent. No matter how good a story we have concocted with respect to what we expect to happen, no matter how careful our analysis, stuff happens that can and often does mess up and mess with our hopes, dreams, and schemes.
As the great playwright Tom Stoppard understood: “[T]here is really, really good news if you end up feeling lucky rather than clever.”
As Stoppard postulated in his play, The Hard Problem: “In theory, the market is a stream of rational acts by self-interested people; so risk ought to be computable. But every now and then, the market’s behavior becomes irrational, as though it’s gone mad, or fallen in love. It doesn’t compute. It’s only computers compute.”
It doesn’t compute, of course, because markets are driven by people, who may be self-interested, but who aren’t necessarily rationally self-interested at any given time. They are mere human people, not computer people. They are yearning, hurting, illogical, infuriating – driven by the attraction Newton left out and by the devout hope that the better angels of our nature that Lincoln saw are real. We want binary questions and answers in a world far messier and more complicated than that.
As Nobel laureate Robert Shiller has clearly explained, “The U.S. stock market ups and downs over the past century have made virtually no sense ex post.”
Let’s stipulate that forecasting the future is always difficult and that predicting how humans will perform under stress is incredibly difficult. By most models, even the very best team in the tournament has only something like a 15-20 percent chance of winning it all. That’s why your office pool can be won by Janice in Accounting who has never seen a game and picks her winners based upon team colors and mascots.
In related news, we shouldn’t be surprised when the market remains “irrational” far longer than seems possible. But we are.
Bottom line: The world is messier, more random, less predictable, and less linear than we assume.
That’s great for the NCAA Tournament but bad for your bracket. It’s also more than a bit problematic elsewhere.
Busted Bracket
As shown above, the highly unlikely happens surprisingly often. In 1991, UNLV was the defending champion and came into the Final Four unbeaten and unchallenged. In the national semi-final, the Runnin’ Rebels met my Duke Blue Devils, a team UNLV had destroyed the previous year, 103-73, in the most lopsided championship game in NCAA Tournament history. It wasn’t as close as the final score might suggest.
Really.
It was U-G-L-Y in 1990 and nearly everyone expected more of the same in 1991. Happily, that wasn’t what happened. The highly unlikely happens surprisingly often.
This story relates to that game but isn’t about the game itself or even the tournament, exactly. In those days, Wall Street trading houses had big tournament pools that featured high entry fees with serious bragging rights and big prizes at stake. Significantly, because there were lots of traders involved, lots of trading went on. You could call almost any major shop and get two-sided markets on pretty much any aspect of the tournament including, of course, which team would win it all.
That final fact is noteworthy because one particular trader was sure that UNLV was going to repeat as champions. More particularly, he was absolutely convinced that Duke would not win the tournament and shorted the Blue Devils big without hedging – expecting to profit handsomely when elimination ultimately came. In other words, he was looking to make big money on the trade and not just on the spread.4 My guess is that he was a prop trader wannabe. Significantly, losing would mean not just lost potential profits – he would have to ante up real cash. Probably worse, he would lose face on the trading floor.
As the expression goes, he was picking up nickels in front of a steamroller.
Our poor schlub was pretty nervous on the Monday after the UNLV upset, but Duke still had to beat Kansas that evening (ironically, it was April Fool’s Day) to win the title for the trader to have to cover his shorts. He feigned confidence, of course, but nobody was fooled. When Duke prevailed over the Jayhawks, 72-65, the fool was six figures (plus) in-the-hole.
The trader made good – sheepishly and painfully – but the brass learned a lesson. Thereafter, the big investment banks no longer allowed employees to organize tournament pools and trading on the pools that existed was strictly prohibited. It was even enforced. Rumor had it that this was part of a quiet agreement between regulators and internal compliance officials, who were understandably concerned about what had gone on. Wall Street pools still existed after that, of course, but they were run exclusively on the buy-side. We on the sell-side still played, but it wasn’t the same. There wasn’t any trading that I’m aware of. And that’s a good thing.
Traders are going to trade – on Apple, GameStop, or Duke – and gamblers are going to gamble. More than $15 billion will likely be wagered on the NCAA basketball tournament this year, with “only” a fraction of that bet legally. Without careful oversight, accountability, position limits, and careful hedging, it’s inevitable that people will get themselves in big trouble. That lesson applies to NCAA Tournament pools, to any security you might want to name, and to life.
Totally Worth It
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Star Mets closer Edwin Diaz suffered a season-ending knee injury Wednesday night during the on-field celebration of Puerto Rico’s victory over the Dominican Republic in the World Baseball Classic.
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Benediction
For this week’s benediction, in honor of St. Patrick, here is a stunning version of a Gaelic blessing.
May the road rise to meet you. May the wind be at your back. May the sun shine warm upon your face. May the rain fall softly on your fields. And until we meet again, May you keep safe in the gentle, loving arms of God.
To those of us prone to wander, to those who are broken, to those who flee and fight in fear – which is every last lost one of us – there is a faith that offers hope. And may love have the last word. Now and forever.
Amen.
Thanks for reading.
Issue 146 (March 17, 2023)
A coin flip, with exactly 50/50 odds, requires 67,600 flips to be 99 percent sure it is not a weighted coin. Oh, and the odds aren’t really exactly 50/50.
As much as I love college basketball, I must note that the NCAA is a non-profit organization that generates over $1 billion annually, over 90 percent of which comes from March Madness … and the players remain unpaid.
You can buy insurance to cover someone winning promotions like that.
Interesting side note: One of the better mortgage traders I knew got his start betting sports in the dark ages before national lines and the internet. He’d bet the underdog in the favorite’s city and vice versa, taking advantage of wide disparities in the lines and creating what was effectively a wide bid/ask spread. He won both sides surprisingly often.
Thank you for an excellent entry today.I look forward to your wisdom every week.
Paul
Daytona Beach FL
Reading in the aftermath of the Purdue loss. Great piece, Bob.