Hong Life CalculatorsUnderstanding the rating difference to win probability calculator — concept and principle
This calculator converts a rating difference between two players or teams into the expected score of the stronger side, using the Elo logistic curve on the standard 400-point scale. It also shows how many rating points of advantage correspond to a 75% expectation. Use it to judge how lopsided a match is, or to understand what a rating gap really means.
How it is calculated
- Expected score = 1 ÷ (1 + 10^(−d ÷ 400)) × 100%, where d is your rating minus the opponent's.
- Difference needed for 75% = 400 × log₁₀(3) ≈ 190.85 points.
The inverse formula, d = 400 × log₁₀(E ÷ (1 − E)), turns any expected score E back into a rating difference.
The principle
The Elo model says that the odds of scoring against an opponent are 10^(d/400): every 400 points multiplies the odds by 10. Odds of x mean an expected score of x ÷ (1 + x), which gives the S-shaped logistic curve. The curve is steepest around a zero difference, where each extra point matters most, and flattens at the extremes, where even a big gap cannot push the expectation above 100%. Setting E = 0.75 means odds of 3 to 1, so 10^(d/400) = 3 and d = 400 log₁₀3.
Worked example
- Input: rating difference d = 100.
- 10^(100 ÷ 400) = 10^0.25 ≈ 1.7783, the odds in your favour.
- Expected score = 1.7783 ÷ (1 + 1.7783) = 1 ÷ (1 + 10^(−0.25)) ≈ 64.01%.
- Difference needed for 75% = 400 × 0.47712 ≈ 190.85 points.
| Rating difference (yours − theirs) | Expected score |
|---|---|
| 25 | 53.59% |
| 50 | 57.15% |
| 75 | 60.63% |
| 100 | 64.01% |
| 125 | 67.25% |
| 150 | 70.34% |
| 200 | 75.97% |
How to read the result
Useful reference points on the 400 scale: a difference of 0 gives 50%, about 70 points gives 60%, about 191 points gives 75%, 200 points gives about 76%, and 400 points gives about 91%. The figure is an expected score, so draws count as half a point; in games with many draws, such as top-level chess, the real chance of a decisive win is lower than the number suggests. Other rating systems built on the same idea may use a different scale constant, so a 100-point gap does not mean the same everywhere.
Common mistakes
- Entering the opponent's rating minus yours, which gives the opponent's expected score.
- Treating 64% as a 64% chance of winning when draws are possible.
- Applying the curve to ratings from different pools or systems.
- Expecting the model to account for home advantage, form or injuries; it does not.
Frequently asked questions
How big a rating gap means a 90% favourite?
d = 400 × log₁₀(9) ≈ 382 points gives an expected score of 90%.
Is the curve symmetric?
Yes. If you expect 64.01%, your opponent expects 35.99%; the two always add up to 100%.
Why 400 points?
It is a scaling choice made when the system was designed so that rating numbers had a convenient spread; a different constant would only stretch the same curve horizontally.
Sources: Arpad E. Elo, The Rating of Chessplayers, Past and Present, Arco Publishing, 1978 · FIDE Handbook, FIDE Rating Regulations