Elo and best-of-series probabilities for esports
Convert an Elo gap into a single-game win probability, then into Bo3 and Bo5 series probabilities, plus what the independence assumption hides.
In this guide
Turning an Elo gap into a single-game probability
Elo is a rating system that maps the difference between 2 ratings onto an expected win probability. The transform in this worked protocol is p = 1 / (1 + 10^((RB - RA)/400)), where RA and RB are the 2 ratings. A zero gap gives p = 0.5. A 100-point gap gives p = 0.6400649998028851, which means the higher-rated side wins roughly 64 percent of a single game under this formula.
That formula is a chosen model. It is not a Valve or Riot ranking, and it is not a validated forecast. No event rulebook converts an Elo number into a guaranteed result, so read the applicable current event rulebook in Riot's official Competitive Operations library for competition structure instead of assuming a generic league procedure. Everything below is an illustrative protocol, not an empirical study of any real league.
Bo3 and Bo5 under the independence assumption
The simplest series model treats each game as independent and identically distributed, usually shortened to IID. That means the same p applies to every game and one game does not shift the next. Under IID, a best-of-3, or Bo3, series ends when a side reaches 2 wins and the probability is 3p^2 minus 2p^3. With p = 0.6 that gives 0.648, so a side winning 60 percent of single games takes about 64.8 percent of Bo3 series.
A best-of-5, or Bo5, ends at 3 wins and the IID probability is 10p^3 minus 15p^4 plus 6p^5. With p = 0.6 that gives 0.68256. Longer series stretch the favourite's edge, from 0.6 to 0.648 in Bo3 and then to 0.68256 in Bo5. Those values are exact for the IID model and for nothing else.
| Single-game p | Bo3 | Bo5 |
|---|---|---|
| 0.50 | 0.5000 | 0.5000 |
| 0.55 | 0.5748 | 0.5931 |
| 0.60 | 0.6480 | 0.6826 |
| 0.6400649998 | 0.7046 | 0.7492 |
A heterogeneous Bo3 example
Real series are not always IID because map choice and side selection can change the per-game probability. Picture an illustrative Bo3 where the favourite's per-game chance is 0.7 on one map, 0.4 on another, and 0.5 on a decider. If each game uses its assigned value, the overall series probability under that conditional mapping is 0.55, well below the 0.648 that IID would give for p = 0.6. A concrete reading of that case: with map probabilities 0.7, 0.4 and 0.5, the favourite wins the first 2 in a row with chance 0.7 times 0.4 = 0.28, and the remaining sequences that end 2-1 add up to 0.27, so the total is 0.55.
The gap between 0.648 and 0.55 is the map-selection process talking. Which map is played, who picks it and how sides are assigned can move the per-game probability itself. The 0.55 figure holds only for that assignment; reordering the maps or changing which side bans first changes it. Treat map choice as an explicit input rather than a detail you average away.
| Map | Assigned win probability | Note |
|---|---|---|
| Map A (favourite pick) | 0.70 | Favourite favoured |
| Map B (opponent pick) | 0.40 | Favourite unfavoured |
| Map C (decider) | 0.50 | Even |
| Series outcome | 0.55 | Conditional on this assignment |
What the IID assumption hides
The closed forms above are clean because they ignore 3 effects. Correlation: form, draft and patch reading can link games, so winning game 1 is not necessarily independent of winning game 2, and correlation can move the true series probability away from the IID value, in either direction, depending on the nature of the correlation. Conditional map choice: the pool and pick order make some games easier or harder. Unknown strength: the Elo rating is an estimate with error around it, so the true p sits in a range around the value you plug in.
Because of those effects, an IID Bo3 or Bo5 number is a baseline, not a prediction. It is useful for checking whether a stated series price sits far from the model, but it does not prove the model is right. For a real event, write down the map choices, the rating source and the uncertainty around p before trusting the output.
A worked decision from Elo gap to series price
Suppose a favourite is 100 Elo above the opponent, so p = 0.6400649998028851. IID then gives Bo3 = 0.7046018521779805 and Bo5 = 0.7492062000201936, so the longer series favours the better side more. If a market price for the Bo5 implies less than 0.7492 for that side, the model says the side is underpriced relative to this transform. That comparison is only as good as the Elo input and the IID assumption, and it is not a recommendation to trade.
Now stress a separate heterogeneous example with its own per-map assumptions. If the favourite's per-map probabilities are 0.7, 0.4 and 0.5, the series probability falls to about 0.55, which is below the 0.7046 IID Bo3 estimate. The losing case for a model follower is clear: buying the favourite at an IID-derived price can be wrong when the map pool is unfavourable. Compare the market not only to the IID number but to a range that reflects map choice and rating uncertainty.
Limits and what to check next
This page contains no tested accounts, no market quotes and no claimed gains. The Elo transform, the Bo3 and Bo5 formulas and the heterogeneous 0.55 case are illustrative constructions from an authored model, not validated forecasts. Real competition structure comes from the applicable current event rulebook kept in Riot's official Competitive Operations library, not from a generic invented procedure.
Before using these numbers on a live event, check 3 things: the source and age of the ratings, the actual map-selection rules for that tournament, and whether recent games show correlation rather than independence. To compare an Elo-derived probability with a market price, see the odds page. To test the model against past results, use the backtest page, and for the wider competitive setting start at the esports page.
Sources & verification
Sources checked
PolyZeno. Automated review with DeepSeek V4.1 Flash.