Weighting the Coin: A Theoretical Case for Nomentum

First off I’d like to dedicate this column to Bill Barnwell, writer at Grantland, who has been pushing a nomentum agenda heavily in his columns this NFL season. After “momentum” came up a couple of times in my first Sh*t Announcers Say post, I thought I’d touch upon it a bit more, in a slightly different light. I’m not going to offer a bunch of numbers; plenty of people have already done that.1 No, I’m going to look at momentum theoretically.2

We sports fans have a lot of theories. We love our theories. We have theories for why that $*&%bird referee made a certain call when he did, what enabled Lebron James to finally win a championship (two of them, actually), and how the Fear the Beard movement propelled the Red Sox to their third world series title in ten years. Milorad Cavic and many others have their theories about Michael Phelps’ touch-out in the 2008 100-meter men’s butterfly Olympic final. Some theories even rise to such prominence that they get names, such as Dave Cirilli’s Ewing Theory. Personally, I have theories about which articles I read covering the 49ers during the week will help them play the best on Sunday3, and when they scored twice to take a 20-14 lead over the Saints in the fourth quarter last week, just after a friend had come over for a little bit, I almost begged him not to leave his seat on the couch.4 We see things, and we try to explain them. Conceptually, it’s like a science. We sports fans are just a little more fanatical about it, that’s all.

Unfortunately, that’s the problem. Struggling for objectivity is boring and lame, but it isn’t like science: It is science! When you watch a game and the momentum shifts in your team’s favor, and you go on to win, it sticks in your mind. “The game totally hinged on that play!” we say. “After that, we knew they couldn’t stop us. You could see it.” When you watch a game and the momentum shifts in your team’s favor, but they go on to lose (or the “momentum” shifts back), it’s forgotten or dismissed. “We had something going, but the game got out of hand.” But just because a team’s momentum didn’t come through once doesn’t mean it’s not responsible for all the times they actually won. After all, theoretically it makes sense, right? That team was in the zone! After that play they knew they were going to win. It was a huge confidence boost, and they put the other guys back on their heels.5

Forget sports (just for a second, don’t worry) and think about a coin flip. Say it’s a fair coin, and you flip heads two times in a row. Does the coin have momentum? Is the coin more likely to come up heads on the next flip? You’re smart, you know the answer is no. It’s just a coin! A fair coin, at that. It’ll come up heads 50% of the time and tails 50% of the time. Even if the coin wasn’t fair, there still wouldn’t be any momentum. If you knew a coin flipped heads 75% of the time, but while flipping there was a run of three tails in a row, would you next bet on heads or tails? Heads, of course. It comes up 75% of the time.6 If you think momentum plays a role in a coin flip, you may be beyond help. But if you agree it doesn’t, you’ve got to agree that momentum plays no role in sports as well.

BUT WE’RE TALKING ABOUT REAL PEOPLE, PLAYING SPORTS, NOT A STUPID COIN FLIP! Well…

In statistics (like, 101, don’t worry, it’s super simple) we say events are either independent or dependent. Coin flips are independent of each other. The outcome of one event does not change the probabilities of the outcomes of any of the others. Two events that would be dependent are whether I wear a rain jacket and whether or not it is raining. The probability that I wear a rain jacket increases dramatically when it is raining.

So sports. Does the outcome of a certain play, or game, or season, change the probabilities of the outcomes of other plays, or games, or seasons? The 49ers just gained a first down. Are they now more (or less) likely to gain another first down than they were before gaining the original first down? Or, the 49ers just won three games. Are they now more (or less) likely to win the fourth game than they would be otherwise? No and no. No! This is not to say the 49ers (and their opponents) are static. The 49ers may have figured out the other team’s defense. That would improve their chances of gaining a first down. They may have gotten better at playing football. That would improve their chances of winning. Those improvements may be reflected in outcomes (gaining a first down, winning a game), as outcomes are certainly dependent on those improvements. But those improvements are not “momentum”! Future outcomes are not the product of prior ones; they’re a product of what the team is doing. The outcomes themselves are independent of one another.

BUT, you say, WHAT THE TEAM IS DOING DEPENDS ON THEIR PREVIOUS OUTCOMES! Sure, teams respond to what’s happened, and may change their strategy, use different players, employ different techniques, etc. And that’s exactly my point. It’s that process, of evaluating performance and making changes, that drives outcomes.

*Hey, over here! Say we have a fair coin that flips heads 50% of the time. We’re having a coin flipping contest (whoo!) and at halftime, we switch to a weighted coin that flips heads 75% of the time. In the second half we flip heads twice in a row. Are we more likely to flip heads on our third flip than we were on our flips in the first half? Yes. Is it because we flipped heads the last two times? No. It’s because this coin comes up heads 25% more often. The probability of flipping heads was 50%, but now it’s 75%, and we’re seeing the difference.

*Say the 49ers offense scores a touchdown against the Rams (whom we7 play this weekend) 50% of the time. At the end of the first half Robert Quinn is injured, also the 49ers have discovered Alec Ogletree and JoLonn Dunbar get out of position on screen passes. These changes improve the 49ers chances of scoring a touchdown to 75%. They come out of half time and score touchdowns on their first two possessions. Are the 49ers more likely to score on their third possession than they were on their possessions in the first half? Yes. Is it because they scored on their last two possessions? No. It’s because Robert Quinn is injured and they are now exploiting Ogletree’s and Dunbar’s weaknesses! The probability of scoring a touchdown was 50%, but now it’s 75%, and we’re seeing the difference.

That’s all football is. Theoretically, that’s all sports are: a coin flip, and the weight of the coin is always changing. Players practice (and take performance enhancing drugs) to weight the coin in their favor. Head coaches spend hours reviewing tape and game planning to weight the coin in their favor. Peyton Manning makes adjustments at the line of scrimmage to weight the coin in his favor. There are no guarantees, and there are a lot of coins. The probability of Dustin Pedroia getting on base coin. The Lebron James free throw coin. The Landon Donovan penalty kick coin. The Jim Harbaugh challenge coin. There are a lot of coins, and they all contribute to The Coin, the probability of winning coin. It could be weighted heavily towards your opponent, as was the 2008 Detroit Lions coin, or strongly in your favor, as was the 2007 New England Patriots coin. But no matter what you do (as those same Patriots would be sure to remind you), the outcome of The Coin is never 100% certain.

So the game starts and maybe the 49ers will score a touchdown 30% of the time against the Rams. After one drive the Rams change their coverage, weighting the coin in their favor, down to 28%. The 49ers try a new wrinkle, weighting the coin in their favor to 31%. Vernon Davis misses a couple drives with a cramp, weighting the coin down to 22%. So it goes. On and on. And if you think looking at the numbers like that takes the fun out of sports, get some glasses, because you’re seeing sports all wrong. Those changes– the freak occurrences, the constant adjustments in preparation and strategy– are what make sports great. And fun. Win or lose.


  1. Again, to get you started, check out the Hot-Hand fallacy. Also a more friendly Bill Barnwell Grantland piece. And an even friendlier New York Times piece
  2. Jeez, it’s almost like I went to a college where a bunch of people wore “That’s all well and good in practice… but how does it work in theory?” t-shirts. Oh wait, I did! Long live Thompson House, and our blessed cake business! 
  3. Surely reading all of Matt Maiocco’s content before kickoff demonstrates my love as a fan, and will be rewarded? Though actually, I didn’t get to all of it before they beat Washington on Monday. But whatever. 
  4. He did, and the 49ers never scored again, losing 23-20. SEE WHAT I HAVE TO PUT UP WITH? 
  5. They made a statement! Changed the complexion of the game. Dictated the game. Took the driver’s seat. Could smell blood.  Answered the call. Started to make some noise. Fired on all cylinders. Hit their stride. Hit a turning point. Turned the corner. Turned the tide. Set the tone. Raised the bar. Played with swagger. Played with a sense of urgency. Were on a mission. Were off to the races. Made a stand up play. Made a gutsy play. Made a textbook play. I could go on… 
  6. An obligatory link to the Gambler’s Fallacy. Three tails in a row doesn’t make heads any more likely either. 
  7. Yes, I say “we” frequently when talking about the team I root for, in this case the San Francisco 49ers. My San Francisco 49ers. Who are (obviously) Jed York’s San Francisco 49ers. Get over it. 
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