The Octagon Is a One-Trial Experiment
Jon Jones was, by almost any measure, the most complete mixed martial artist ever assembled. In 2023 he returned from a three-year layoff to fight Ciryl Gane for the heavyweight championship. Jones won by first-round submission. It was not particularly close.
Now imagine you had never seen Jones fight. You watched exactly one round of that contest and were asked: what is the probability Jones wins this match?
If you’re a frequentist, you have a problem. Frequency-based probability is formally defined as a limiting ratio — the number of times an event occurs divided by the number of opportunities, as the number of opportunities goes to infinity. A single fight is one trial. You cannot take a limit over one data point. So strictly speaking, the frequentist cannot assign a probability to this specific contest.
This is not a minor inconvenience. It is one of the oldest and most discussed objections in the philosophy of probability: the single-case problem, treated at length by Hájek (2009) and Gillies (2000).
But here is the thing: most people watching that fight thought Jones was a heavy favorite, and they were right. And they were right in a way that a frequentist framework, properly applied, can actually explain — just not in the way the objection implies.
The standard critique goes like this: frequentism requires repeatable experiments. A unique event — this fight, this night, with this version of these two people — can’t be repeated. Therefore frequentism is useless for one-off events. Bayesians, by contrast, treat probability as a degree of belief updated on evidence, which works fine for singular events.
The critique is not wrong as far as it goes. But it conflates two things: the probability of a fight type and the probability of this specific fight.
When a model estimates that Jones wins a matchup like his bout with Gane with, say, a 70% probability, it is not claiming something about a unique unrepeatable event. It is claiming something about a class of contests — fights between a wrestler of Jones’s profile and a striker of Gane’s profile, under roughly these conditions. The probability is a statement about the reference class, not the individual.
This is exactly von Mises’s original framing: probability is always about collectives, not single instances. What we need is the right collective — and choosing that collective is the hard part.
The empirical floor: what the betting market knows
Betting markets in MMA aggregate the predictions of many informed people, and their track record gives us a useful empirical anchor.
Across all UFC fights, favorites win somewhere in the range of 60 to 65 percent of the time — meaning underdogs pull off upsets in more than a third of all fights. That 35-to-40 percent upset rate is notably higher than in most major team sports.
Heavy favorites do better: UFC odds priced between -400 and -900 (implying an 80-to-90 percent win probability) have returned an 88 to 93 percent win rate since 2013. At the other end, fights where odds land between +100 and -122 — roughly a coin flip — have resolved in favor of the nominal favorite only 51 percent of the time.
The market, in other words, does predict fight outcomes with meaningful accuracy when there is a genuine skill gap. It fails — or at least flattens — when the gap is small.
Skill is real; one fight is still noisy
Here is the central claim, stated plainly: a fighter being more skilled than their opponent does not mean they win every fight. It means their probability of winning a single fight is greater than 0.5, perhaps substantially so. But a 70% win probability is also a 30% loss probability.
This is not a trivial point. It is, in miniature, the same confusion that afflicts applied researchers who treat a statistically meaningful estimate as proof that the effect always occurs. The estimate is about the distribution. The distribution is not the single draw.
When Georges St-Pierre was 170 pounds of organized violence in his prime, his aggregate probability of winning any given welterweight fight was very high. But any single fight carried:
- Opponent stylistic advantages that a model underweights
- Chance of injury at any moment
- Variance in execution (did the game plan hold? did the cuts go as expected?)
- Pure stochastic noise — scrambles in grappling are notoriously non-deterministic
A single punch, a well-timed submission, or a referee stoppage can override any statistical edge a fighter might carry on paper. A scramble on the mat can convert a dominant position into a finished fight in one second.
If you ran that version of GSP against a solid but beatable opponent ten times, you’d expect him to win eight or nine. In one fight, you’d expect him to win — but “expect” means something probabilistic, not certain.
The reference-class problem is the real fight
Choosing the right reference class is where frequentist analysis actually gets difficult in MMA, and where most naive applications go wrong.
Holmes, McHale, and Żychaluk (2023) proposed a Markov chain simulation approach that explicitly estimates fighters’ skills in key aspects of the sport — striking, grappling, clinch work, etc. — and then simulates the fight rather than simply predicting a binary outcome. This matters because the outcome of a fight is not a single draw from a bivariate distribution. It is the aggregate result of hundreds of micro-decisions, each of which has its own stochastic component.
When you model it that way — as they did, comparing favorably to bookmaker odds — you are treating the fight as a sequence of probabilistic events from a well-defined class. The frequentist framework is doing real work. It is just applied at the level of techniques and positions rather than whole fights.
That is a defensible and scientifically useful construction. It is also, notably, the same move statisticians make in clinical medicine: you cannot run a randomized trial on a specific patient, but you can estimate their probability of response from a well-defined reference population and then apply that estimate to guide the decision.
What this means for how you talk about fight outcomes
The mistake most analysts — and most fans — make is binary: the favorite “should” win, the underdog “shouldn’t,” and an upset is a surprise requiring explanation.
The better frame: a fight between a 70% favorite and a 30% underdog is not a fight with a predetermined outcome. It is a draw from a distribution with that shape. Three out of ten times, the underdog wins. That is not an upset requiring special explanation. That is the distribution working correctly.
An upset does warrant explanation if it happens at a rate higher than the odds imply — i.e., if the market is miscalibrated. The Fightful analysis of heavy favorites suggests the market is reasonably calibrated at the extremes, noisier in the middle.
The key methodological point: a single fight is a sample of size one from a fighter’s underlying skill distribution. Skilled fighters have higher-mean distributions with less variance; fighters with fight-ending power but defensive liabilities have wide distributions. The outcome of one draw tells you something, but not much.
This is why it is statistically incoherent to say, after a single upset, that the underdog was “always better.” It is equally incoherent to say, after a dominant win, that the favorite “always wins this.” One draw is one draw.
The Bayesian objection, and why it doesn’t fully resolve things
Bayesians handle this better in some respects. They assign a prior belief about the outcome, update on evidence, and end up with a posterior. For singular events, this is the most coherent framework — a posterior probability genuinely represents a degree of belief about this specific contest, not a reference class.
But the Bayesian approach faces its own problem in combat sports: you need a prior, and priors in MMA are hard to specify. Two fighters may never have faced similar opponents, may have competed in different promotions, and may have trained specifically for this contest in ways the market cannot observe.
In practice, the most useful approach is not doctrinal but instrumental: build the best reference class you can using comparable fights and fighters, fit a model, and interpret the output as a probability statement about the type of contest this represents, while acknowledging that any single fight is an unreliable single observation of the underlying process.
That is frequentism in practice — not a claim that probabilities don’t apply to single fights, but a demand that you ground the probability in something empirically defined.
The upshot
MMA is genuinely less predictable than most sports. The upset rate is substantially higher than in major team sports, and the mechanisms are clear: it is a sport defined by rapid phase transitions (standup, clinch, ground), each carrying its own variance.
That unpredictability is not a bug in the analysis. It is a feature of the process. A skilled fighter’s edge is real and measurable in the aggregate — the betting market shows this plainly at heavy price differentials — but it is not deterministic in a single contest.
The fighter who “should” win is the one with the higher probability under the best available model. Whether they win this time is a separate question, and anyone who collapses the two has made the same mistake researchers make when they treat a significant p-value as proof that the effect always occurs.
One fight is one trial. The distribution is real. The single draw tells you much less than you think.
See also: - Semantic and Cognitive Tools to Aid Statistical Science — the case for replacing “significant” with “compatible” and describing what the data are more and less consistent with - [TODO: link to another relevant post on inference under uncertainty]
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