From a single game to a series
Suppose Team A beats Team B in any single game with probability p, and the games are independent. In a best-of-1 the favourite wins with probability p, and that is the whole story. In a longer series the favourite needs to win a majority of games, which means chance has to cooperate several times for the underdog to come through.
The effect is a form of the law of large numbers: more games means the observed result drifts closer to the underlying strength. It does not eliminate upsets. It only makes them rarer.
The formulas
A best-of-3 is won by the first side to reach 2 games, a best-of-5 by the first to 3, and a best-of-7 by the first to 4. Write q for 1 - p, the underdog’s per-game chance. The favourite’s series win probability is the sum over every way it can reach the target before the opponent does:
- Best-of-3: p^2 (1 + 2q)
- Best-of-5: p^3 (1 + 3q + 6q^2)
- Best-of-7: p^4 (1 + 4q + 10q^2 + 20q^3)
The numbers inside the brackets count the orderings in which the favourite can win while the underdog collects 0, 1, 2 or 3 games. They come from binomial outcomes, the same counting used for any repeated trial, and the coefficients are binomial coefficients: 1, 2 for the best-of-3, then 1, 3, 6, and 1, 4, 10, 20.
A worked example at p = 0.60
Take a favourite who wins 60% of single games, so q = 0.40.
- Best-of-3: 0.36 x (1 + 0.8) = 0.36 x 1.8 = 0.648
- Best-of-5: 0.216 x (1 + 1.2 + 0.96) = 0.216 x 3.16 = 0.6826
- Best-of-7: 0.1296 x (1 + 1.6 + 1.6 + 1.28) = 0.1296 x 5.48 = 0.7102
A second case at p = 0.55
For a smaller edge, q = 0.45.
- Best-of-3: 0.3025 x 1.9 = 0.5748
- Best-of-5: 0.166375 x (1 + 1.35 + 1.215) = 0.166375 x 3.565 = 0.5931
- Best-of-7: 0.09150625 x (1 + 1.8 + 2.025 + 1.8225) = 0.09150625 x 6.6475 = 0.6083
Reading the table
| Per-game chance | Best-of-1 | Best-of-3 | Best-of-5 | Best-of-7 |
|---|---|---|---|---|
| 0.60 | 0.600 | 0.648 | 0.683 | 0.710 |
| 0.55 | 0.550 | 0.575 | 0.593 | 0.608 |
Equivalently, the underdog’s upset chance at p = 0.60 falls from 40.0% in one game to about 35.2%, 31.7% and 29.0% across the longer formats.
Two things stand out. First, the gains shrink as the series grows: going from one game to three adds more than going from five to seven. Second, a small edge stays small. A side that is only a 55% favourite per game is still beaten roughly four times in ten in a best-of-7. Series length amplifies an advantage but cannot create one that is not there.
Why the real world is messier
The tidy numbers rest on assumptions that rarely hold exactly.
- Independence. Games in a series are connected. Teams adapt, morale shifts, and fatigue can build up. Treating each game as one of several independent events is a simplification.
- Fixed per-game probability. Map or character picks, side selection and veto systems mean the true chance varies from game to game. A team strong on one map and weak on another has a different p each time.
- Early stopping. The formulas assume the series ends as soon as someone clinches it. Playing out all the games gives the same series winner, which is why the sums above can also be built from a fixed number of games.
- Uncertain p. The per-game probability is itself an estimate. If it comes from an Elo rating, a 100 point gap, for example, implies roughly 0.64, and the estimate is only as reliable as the rating behind it.
There is also a practical cost. Longer series take more time, and organisers have to balance fairness against schedules. Early rounds are often shorter and finals longer, which is a trade-off between reducing luck where it matters most and keeping a tournament feasible.
A quick check for any format
To judge a new format, find or estimate p, plug it into the formulas above, and compare the result with the shorter alternative. If the difference is only a percentage point or two, the extra games are mostly adding length, not accuracy. If the teams are closely matched, the series remains close to a coin flip however long it runs.