Two forces in every result
A match result is shaped by two things: what the players chose, and what the dice did. Choices tend to push results in a consistent direction. Chance pushes them in an unpredictable one. Measuring the impact of randomness means comparing the size of those two pushes.
Statistically, a player’s choices shift the average outcome, and randomness sets the spread around that average. If the shift is large relative to the spread, the better choice wins most of the time. If the spread is large relative to the shift, results look close to a coin flip even when one side is clearly better.
A damage race, step by step
Take a simple duel. Each player makes 10 independent attacks. Every attack has a 20 percent chance to crit for double damage. Player B deals 100 base damage per attack. Player A deals 110, a 10 percent advantage, which stands in for better choices such as stronger positioning, a better build or a better target.
Without crits. Damage is fixed. A deals 1,100 and B deals 1,000, and A wins every time. Randomness has no impact, and the whole result comes from choices.
With crits. Each attack averages 1.2 times its base damage, and the standard deviation of one attack is 0.4 times its base (a base of 100 gives an average of 120 and a standard deviation of 40, as worked out in the discussion of critical hits).
Over 10 attacks:
- B: average 1,200, standard deviation 40 × √10, about 126.
- A: average 1,320, standard deviation 44 × √10, about 139.
The difference between the totals has an average of 120. Its standard deviation combines both spreads: the square root of (139² + 126²), which is about 188. So A’s lead is 120 on average with a spread of around 188. Using a normal approximation, which is rough for only 10 attacks, A comes out ahead roughly 74 percent of the time.
So a clear 10 percent advantage, which wins 100 percent of the time without randomness, wins around three games in four once crits are in play. About one game in four goes to the weaker side purely through dice.
With more attacks. Over 100 attacks each:
- Difference in averages: 12,000 versus 13,200, a gap of 1,200.
- Standard deviation of the difference: the square root of (440² + 400²), about 595.
The gap is now roughly two standard deviations, and A wins around 98 percent of the time under the same approximation. The advantage and the randomness are the same per attack. Only the number of trials changed.
This is the central lesson: the impact of chance on a result depends strongly on how many random events are combined. Few events mean chance has a large say. Many events let a consistent edge show through.
Signal and noise
The calculation above can be summarised as a ratio. The signal is the average advantage from choices. The noise is the standard deviation of the result. When the ratio is below about one, the weaker side wins often. At two, the stronger side wins in the great majority of games. The noise grows with the square root of the number of events, while a per-event advantage adds up in direct proportion, so the ratio improves with the square root of the length of the contest.
This helps explain why games with few, high-impact random events feel luck-driven and games with many small ones feel skill-driven, even when the average random effect per event is identical.
Features that raise or lower the impact of chance
Several design and play factors change how much randomness matters:
- Size of the swing. A rare, large effect adds far more spread than a frequent, small one with the same average. This is the difference between a 25 percent chance of double damage and a 50 percent chance of 1.5 times damage.
- Number of events. More rolls dilute any single roll. Longer matches and longer series favour the stronger side, which is the reason series length is a design lever.
- Timing of randomness. Chance that resolves before a decision lets the player respond. Chance that resolves after one cannot be answered.
- Comeback and snowball mechanics. If an early lucky event compounds, a small random difference can turn into a large one. If the game pulls trailing players back, early luck matters less.
- Player control over exposure. Good players can sometimes choose lower-variance or higher-variance options depending on whether they are ahead or behind. This is where choices interact with randomness, and it links to ideas in risk management.
How to estimate impact in practice
You rarely get a clean model, but a rough estimate is often possible.
- Look at how often the side with the higher skill rating loses in your game. If favourites lose around a third of the time even with large rating gaps, randomness is large.
- Compare results as the sample size grows. If longer matches or series favour the stronger side much more clearly, randomness is playing a visible role in short ones.
- Separate average from spread in the events themselves, as in the crit example above.
- Be careful with a single data source. Win rates in one matchmaking pool, for instance, can understate randomness because ratings already pair similar players.
None of these gives a perfect number, and each depends on assumptions about independence and about what counts as a “choice”. They are best treated as ways to bracket the size of the effect, not to nail it down.
Interpreting the answer
A large impact of chance does not mean skill is irrelevant, and the wider question of how luck and skill share a result is treated separately. In the example, A still wins a clear majority of the time with only a 10 percent edge. A small impact of chance does not mean a game is purely skill either; it may just mean the sample is long. The informative quantity is the ratio of edge to spread, because that is what determines how often the better side wins and how many games you need before the difference shows.
Understanding this ratio also guards against two opposite mistakes: blaming every loss on luck, and crediting every win to skill.