“Well, it was an even chance . . . if my calculations are correct.”
— Guildenstern, Rosencrantz and Guildenstern Are Dead

This piece is off to a hot start. A featured image making a Russian roulette/gambling joke and a keystone quotation from a play about two guys written into existence just to die: Could you ask for more?

As much as any piece of mine is about anything, this piece is about the gambler’s fallacy and daily fantasy sports.

What are the Odds?

Random question: If you asked 100 people to name their favorite Shakespeare play, what would the responses be and in what proportions?

The responses and percentages might look like this:

  • 85 percent – The respondent names Hamlet.
  • Seven percent – The respondent names another play.
  • Four percent – “I don’t have a favorite.”
  • Four percent – “I don’t know any Shakespeare plays.”

In this hypothetical example, I think the last three responses would be sincere. For instance, my favorite Shakespeare play is King Lear, not Hamlet. (Bold, I know.) There have to be some other people out there like me, right? And what incentive would people have to admit that they don’t know any Shakespeare plays or that they don’t know any of them well enough to say which one is their favorite?

But what about the 85 percent who say Hamlet is their favorite? In this scenario, if someone names Hamlet, what are the odds the person has actually read the play? — or has read another Shakespeare play against which they can compare it and say, “Yep, that’s my favorite”? I think the odds would be embarrassingly low.

Where am I going with this? I’m not sure. I think my larger point is something like “Statistics and outcomes don’t always mean what we think they mean.” That sounds about right.

Really I was just looking for an excuse to reveal that Lear‘s my favorite play. That says a lot about me . . . or maybe not. (If you ever meet someone who says his/her favorite play is Titus Andronicus, run. There are non-zero odds [s]he is at best a serial killer.)

Rosencrantz and Guildenstern Are Dead

In 1966, playwright Tom Stoppard went from ‘guy with potential’ to ‘superstar writer’ when his Tony-Winning play Rosencrantz and Guildenstern Are Dead was first performed at the Edinburgh Festival by the Oxford Theatre Group.

For those of you in the 85 percent who favor Hamlet: R&G are bit characters in Shakespeare’s play. They’re old friends of Hamlet’s, but eventually they spy on him and attempt (unsuccessfully and to their detriment) to manipulate him. Perhaps because of their betrayal, Hamlet forges a document ordering their execution, which they (presumably unaware of the letter’s contents) deliver to the English king. At the end of the play, news comes to Denmark that “Rosencrantz and Guildenstern are dead.”

Basically, R&G are created to be killed. In a play with many round characters, they are flatter than my desk. In a presciently proto-postmodern move, Stoppard wrote a play in which R&G are the protagonists. In 1990 the play was adapted into a movie, the beginning of which is (sort of) famous:

Naturally, I have some thoughts:

  1. Guildenstern is right (even if he’s making a false logical connection): He and Rosencrantz are within un-, sub-, or supernatural forces. The laws of probability don’t apply to them.
  2. By feeling so certain that he should win at least one of the coin flips, Guildenstern is committing the gambler’s fallacy . . .

. . . maybe.

The Gambler’s Fallacy

One of my New Year’s resolutions was to read more, as I think that reading is winning in DFS. Also, reading is one of Justin Phan’s micro habits, so it’s probably not a horrible idea.

My wife has accepted that I write about random stuff for a living, so sometimes she’ll buy books for me that basically seem like Jeopardy prep courses. I read them — because that’s what I do — and I occasionally come across topics that interest me.

For instance, she recently gave me a book called The Little Book of Answers. (You don’t need to read it.) In the book are little blurbs consisting of questions and answers that collectively impart information likely to be useful perhaps only if one plays trivia regularly.

Here’s one such blurb (and it actually might be the worst one in the book):

If a coin is tossed and lands tails ten times in a row, what are the odds that it will be heads on the eleventh try?

After a coin has been tossed and landed tails ten times in a row, many amateur gamblers would be inclined to bet that the “law of averages” would favor the coin landing heads on the eleventh try. The problem is, the law of averages doesn’t exist. The coin’s probability of landing heads is still fifty-fifty — the same as on each previous toss.

Clearly, that makes good intuitive sense — except maybe it doesn’t.

Sometimes 50/50 Isn’t Really 50/50

I don’t remember if he mentioned this on a podcast or an old RotoViz email thread or a super private and confidential conversation, but when FantasyLabs Co-Founder Jonathan Bales was younger he used to exploit bad prop bets offered by online sportsbooks. (On the “Vegas” episode of The DFS Roundtable with Bales, Sean Koerner, and Rob Pizzola there’s some talk about prop bets near the end.)

Specifically, Bales would bet as much money as he could on particular opening kickoff bets: Which team would receive the ball to start the game? While the books were treating each of these bets as 50/50 propositions, Bales noticed that many teams had particular tendencies. Some of them would almost always elect to receive if they won the opening coin toss, and others would almost always elect to kick. When these two types of teams played against each other, Bales would bet accordingly. Eventually, the online books stopped letting him make those bets.

Not everything that looks like a 50/50 proposition actually has 50/50 odds.

The Prior Bayesian Prior

A few years ago Evelyn Lamb (a freelance math and science writer) published a Scientific American blog post about the (im)probability of R&G flipping coins and getting heads 76 (and als0 90) times in a row. She said this:

Wolfram Alpha tells me that 276 is about 7.5×1022, so the chances of getting 76 heads in a row is about 1.3×10-23, which is much closer to 0 than it is to 1 in 7 billion. Even if everyone in the world has flipped coins at least 76 times, there’s almost no chance that anyone’s last 76 flips have all been heads.

So what is the longest string of heads that it’s likely that anyone in the world has seen, assuming that they are flipping fair coins that have a 50/50 chance of coming up heads on any individual flip?

For me, the question isn’t how many times someone is likely to have flipped heads on the way to the world record. Rather this question is much more fascinating (and probably relevant): How many consecutive heads (or tails) in a row need to be flipped before the odds are greater than not that some sort of manipulation is occurring?

This inquiry has a Bayesian perspective: Given the prior probability that no manipulation is taking place, what are the updated odds with each consecutive heads (or tails)?

I’ve written before about the Bayesian prior: Given that the Warriors lost the NBA Finals after having a 3-1 lead, that piece looks either brilliant or idiotic in retrospect, depending on your perspective.

Anyway, if I were betting on tails and someone else were flipping the coin, after five straight heads I probably wouldn’t think anything were amiss. At 10, I’d maybe think that I were witnessing a pretty cool and totally random streak that just happened to be going against me. At 11, I’d likely be about 99 percent sure that I were being cheated. Mind you, I’m not a professional (or even amateur) mathematician.

Maybe it would be 15 heads instead of 11. Either way, the point is there would be a threshold at which I would stop thinking, “What are the odds of this happening?” and start thinking, “What are the odds that this isn’t luck? — and what are the odds that this will continue to happen?”

If the coin flip were a 50/50 proposition, it would be just as stupid for me to think that the streak of tails would continue as it would be for me to assume that the streak of non-heads must end. It would be sort of the bizarro version of the gambler’s fallacy. At the same time, if I ran into a streak of 76 heads it would be wise for me to question whether I were truly playing a game of chance.


Although luck is always involved in everything, DFS and the professional sports on which its based are games of skill.

Sometimes DFS players have a gambler’s fallacy perspective. As I write this it’s March 10, 2017. If I go to our Player Models and look at FanDuel shooting guards for the main slate, I see Seth Curry has a 100 percent Consistency Rating over his last 10 games:

If I were looking at Curry the Lesser from a gambler’s fallacy perspective, I would likely assume that his streak of DFS value were likely to end soon, maybe even in this slate.

Of course, such a perspective would likely not take into account that his +6.09 Plus/Minus over the previous 10 games suggests that Curry, despite his $700 increase in salary, still affords a wide margin of error as a DFS investment — especially given his 95 percent Bargain Rating, top-three +2.97 Opponent Plus/Minus, and slate-high Pace Differential.

Quick aside: I’m not someone to listen to for NBA DFS advice, but . . .

  1. I’m using Curry as an example. I’m not saying that you should actually roster him this slate or that he’s a stone-cold lock to meet value.
  2. Bryan Mears did analyze Curry in today’s NBA Breakdown. If you want to know how Curry fits into the dynamics of the slate, see his piece.

The big point is that, like fallacious gamblers, a lot of people throw around words like “regression” and “reversion” without taking the time to think about the odds or do the necessary research.

Sometimes players and teams over- or underperform, and it’s likely that their output will change. Other times, though, trends hold. (To see whether a trend is worth something or nothing, Pro subscribers can research for themselves via our Trends tool.)

If you’re a turkey, it’s possible that Thanksgiving is approaching and you’ll soon be dead (per Nassim Nicholas Taleb’s Black Swan). Reversion can be sudden and horrible. Of course, to know if you’ll die soon you first need to learn whether you live in a place that celebrates Thanksgiving — because most of the world doesn’t. You need to research.

Super Bowl 51, Weeks Later

During the 2016 NFL season, many DFS players and bett0rs kept waiting for the Falcons to regress. They didn’t. As I highlighted in my masterpiece bowl game breakdown, the Falcons regularly hit their implied point totals throughout the season. Similarly, a number of people expected the Patriots to become less dominant as the season progressed. That didn’t happen.

Entering Super Bowl 51, the Patriots were 15-3 against the spread, and Falcons games were 15-2-1 on the over. Is it a surprise that the Patriots covered the spread against the Falcons in a game that hit the over?

The gambler’s fallacy is idiotic enough in a game of chance. In a game of skill it’s probably even worse.

The Labyrinthian: 2017.23, 118

This is the 118th installment of The Labyrinthian, a series dedicated to exploring random fields of knowledge in order to give you unordinary theoretical, philosophical, strategic, and/or often rambling guidance on daily fantasy sports. Consult the introductory piece to the series for further explanation. Previous installments of The Labyrinthian can be accessed via my author page.