Suppose you have a decent model for a single game вАФ say a Poisson model that, given two teams, spits out the probability of a home win, a draw, and an away win. Now someone asks the bigger question: "What's the probability this team lifts the trophy?" It's tempting to reach for a calculator and start multiplying. Resist that instinct. For anything past the simplest bracket, hand-multiplication quietly falls apart, and Monte Carlo simulation is the tool that actually works.
This article explains how to go from a match-level model to tournament-level probabilities by simulating the whole event thousands of times вАФ how the loop works, how many runs you need, how to attach confidence intervals, and where the approach can mislead you.
Why multiplying probabilities by hand breaks down
Picture a knockout bracket. For a team to win, it has to survive the round of 16, the quarter-final, the semi, and the final. If those four matches had fixed opponents and fixed win probabilities, you could just multiply: 0.7 √Ч 0.6 √Ч 0.55 √Ч 0.5 and be done.
The problem is that the opponents are not fixed. Who your team meets in the quarter-final depends on who won the other round-of-16 tie вАФ which is itself uncertain. To do this by hand you'd have to enumerate every possible path through the bracket, weight each path by the probability that this exact set of results occurred, compute your team's chance along that specific path, and sum over all of them. The number of paths explodes combinatorially. Add group stages with tie-breakers, seeding, re-seeding, byes, or extra-time-then-penalties, and the bookkeeping becomes hopeless.
