Module 01 taught that a YES price on a $1 binary is read as implied probability. Module 02 showed you rarely trade at that mid—you pay the ask and sell at the bid. Module 03 explained how arbitrage keeps related prices coherent when payoffs truly match. This chapter gives the formal language underneath all three: what probability means, which rules every coherent belief must obey, and how to sanity-check market quotes before you claim edge.
Outcomes, events, and sample space
An experiment is waiting for uncertainty to resolve—election night, a CPI print, a court ruling. The sample space Ω is the set of mutually exclusive, exhaustive outcomes. For a binary Kalshi contract, Ω might be {YES resolves, NO resolves}. For a three-way nomination market, Ω might be {Candidate A wins, B wins, C wins}. For “rate cut by March,” Ω is often {cut happens, cut does not}.
An event A is a subset of Ω you care about. “YES wins” is one event; “margin greater than three points” might be another on the same night. Do not mix them when you read prices. Two contracts on the same headline can have different Ω if resolution text differs—that is why arbitrage sometimes fails despite similar titles.
The Kolmogorov axioms
Any assignment of chances must satisfy three rules.
Non-negativity: P(A) ≥ 0 for every event A. You cannot assign negative chance to an outcome.
Normalization: P(Ω) = 1. Something on the menu happens; probabilities across an exhaustive list sum to 100%.
Additivity: if A and B are mutually exclusive (both cannot happen), then P(A ∪ B) = P(A) + P(B). “Wins Iowa OR wins New Hampshire” only adds cleanly if both cannot occur together.
Violating these internally means your book is incoherent. Arbitrageurs can trade against you the same way they trade mispriced sibling contracts—your beliefs contradict themselves, so someone can construct a package that profits at your expense when settlement rules align.
Derived rules you use daily
From the axioms alone you get useful shortcuts.
The complement rule: P(not A) = 1 − P(A). YES versus NO on a true binary is the everyday case.
Monotonicity: if event A implies event B (every world where A happens, B happens), then P(A) ≤ P(B). “Wins the presidency” cannot be less likely than “wins Pennsylvania” if a Pennsylvania win is genuinely nested in the same resolution tree. If your numbers say otherwise, check the contract text, not just the headline.
Bounds: 0 ≤ P(A) ≤ 1. Reject quotes above $1 or negative prices before fees.
When events overlap, the general union formula applies: P(A or B) = P(A) + P(B) − P(A and B). You subtract the overlap once so you do not double-count.
Binary complement in prose
A Kalshi-style “Fed cuts 25 bp at March meeting” shows YES ask at 36¢. Your buy-side implied probability is 0.36; the complement NO is 0.64 if NO is a perfect mirror. If NO ask is 66¢, the two executable sides sum to 1.02. That two-cent overround is spread, fees, and microstructure—not a free two percent. Cross-contract arbitrage only works when rules and payoffs truly match.
Screen shows 61% on “BTC ETF approved by date X” with pool buy 63¢ and sell 59¢. Use 63¢ for a buyer’s axiom check: implied P ≤ 0.63. If NO ask is 40¢, sum 0.63 + 0.40 = 1.03—that is not a violation of probability theory; it is overround from spread economics.
Mutually exclusive trio
In a three-candidate primary, your model might be 42% / 28% / 30%, summing to 1.00. Market mids might read 40% / 30% / 32%, summing to 1.02. A sum above one is common when each leg has bid–ask width. Compare executable prices—asks if you are buying every outcome, bids if you are selling the bundle—not mids alone, before you call structural arbitrage.
If you buy YES on every outcome at the asks—48¢ + 35¢ + 22¢ = $1.05—you own the sample space and receive $1 at resolution. You paid five cents of overround: negative expected value with no view unless individual legs were mispriced.
Overlapping events and incoherence
“Party wins popular vote” and “party wins presidency” are not automatically nested in U.S. history. Resolution text decides. If your tree says P(presidency) = 0.55 but P(popular vote) = 0.48 while you believe presidency implies popular vote, you have violated monotonicity or misread the contracts.
Mini self-check: Ω = {Cut, Hold, Hike}, only one occurs. You believe 50% / 35% / 20%—sum 1.05, incoherent. Normalize to roughly 47.6% / 33.3% / 19.0% or revise judgments. Mids on screen that sum to 1.05 are not automatically free money; check whether buying all three via asks costs more than $1 after fees.
Interpretations
Frequentist probability is long-run frequency; it fits poorly for one-off elections. Bayesian probability is degree of belief given information—the forecaster’s default, developed in the next chapter. Market (operational) probability is the price at which capital clears—what Kalshi and Polymarket display.
All three can coexist: you hold a Bayesian P, the market prints a price, and after many calibrated bets your win rate should line up with stated probabilities if you are honest.
Laplace (“equally likely” ⇒ 1/n) helps for dice, not for elections. The crowd’s price substitutes for symmetry. Treating “no idea” as 50/50 is a claim, not an axiom.
Coherence versus efficiency
Coherence is about your numbers not contradicting. Efficiency is about price matching true frequency. A coherent forecaster can be wrong; an incoherent book is arbed when payoffs align. Market makers on regulated CLOBs and bots on pools work to keep YES/NO near complement; your checklist runs before you claim the crowd is mispriced.
Common mistakes
Using mid as gospel while overstating edge. Adding correlated contracts as if they were disjoint. Ignoring the NO leg when YES at 40¢ and NO at 64¢ both carry friction. Assuming coherence on screen means efficiency. Buying two disjoint primaries on the wrong tree creates resolution risk undefined by the axioms.
Kolmogorov (1933) put probability on measure theory so intuitive rules could not smuggle contradictions. Prediction markets are finite Ω in practice; treat the axioms as accounting identities for belief capital.
Resolution text defines Ω
Before any arithmetic, copy the exchange rulebook sentence that decides YES. “Wins the election” without office, date, or data source is not a well-defined event—it is a headline. Two platforms listing the same news story can have different Ω if one counts runoffs and the other does not. Module 03’s cross-venue gaps are often different experiments, not broken math.
When you list outcomes for a categorical market, ask whether exactly one row will pay $1. If the exchange might void or refund on ambiguity, Ω is larger than the marketing graphic suggests. Probability one applies to the declared resolution set, not to your political intuition alone.
Finite additivity on baskets
Additivity is why “buy every outcome” is a complete story. If there are three mutually exclusive winners and you own YES on each, you own the sample space. Fair cost is $1 per complete set before fees. Paying $1.05 via asks means you donated five cents to liquidity providers and takers—negative EV even with no directional view.
Selling YES on every outcome is the mirror: you are short the sample space. If bids sum to $0.97, someone can buy all YES legs from you and lock three cents before fees. Bots scan those sums because violations are mechanical, not ideological.
When the crowd enforces axioms
You do not need to prove Kolmogorov to trade. You need to notice when your spreadsheet disagrees with executable prices on a shared tree. Market makers on Kalshi-style books lean YES and NO toward complement; Polymarket bots sell when YES plus NO drifts above $1 on equivalent shares. That enforcement is conditional on matching payoff and oracle language.
Your private job is narrower: write Ω from rules, check complements at the ask you would pay, and flag overlaps before you add probabilities from different contracts. The next chapter adds conditioning—updating those numbers when evidence arrives instead of restarting from fifty-fifty.
Keep a one-line Ω in your journal next to every open position. When the sum of asks on a complete set drifts above $1.02, ask whether you are looking at arb or at different products. The axioms are the first filter; execution and resolution are the second.
What comes next
Next: conditional probability and Bayes’ theorem—how to update beliefs when polls drop and prices jump, without confusing “likely given evidence” with “evidence given likely.”