Why displayed chances feel wrong
Players of turn-based tactics and strategy games often report the same thing: shots at 95% seem to miss far too often, and shots at 5% seem to land. Sometimes that points to a genuine quirk in how the number is calculated. More often it comes from how people perceive repeated chances.
This article covers the genuine quirks and the perception problems separately, since the fix for each is different.
Perception first: small chances add up
A 95% chance to hit sounds almost certain, but it still means a 5% chance of a miss each time. Across a whole mission, many shots are taken.
- Chance that 20 independent shots at 95% all hit: 0.95²⁰, about 0.36, or 36%.
- So the chance of at least one miss among 20 shots is about 64%.
In other words, seeing a few misses on high-percentage shots over a long game is the expected result, not evidence of cheating. Similarly, a 5% chance to hit will succeed about once in every 20 tries on average, and over a large number of shots some will land.
There is also a memory effect. A miss at 95% feels dramatic and sticks in the mind, particularly if it costs a unit. The 19 hits before it were unremarkable and are forgotten. This is one reason why people’s impressions of their own luck are poor evidence, a theme that comes up in discussions of risk perception.
Rounding
Displayed percentages are usually rounded. A true chance of 75.4% and a true chance of 74.6% can both show as 75%. The difference is small for a single shot, but it matters in two ways:
- A displayed 100% might not be exactly 100%, and a displayed 0% might not be exactly 0%. Some games clamp the range, and others round tiny values. The user interface may not match the underlying rule at the extremes.
- If several modifiers are each rounded before they are combined, the final figure can differ from what you would get by combining the exact values and rounding at the end.
Rounding rarely causes more than a percentage point or so of error, so it explains very little of the sense that the numbers are wrong, but it is worth knowing about.
Stacked rolls
A single displayed chance sometimes hides more than one check. Suppose an attack first has to hit and then has to get past a block.
- Chance to hit: 80%
- Chance not blocked: 90%
- Chance of dealing damage: 0.8 × 0.9 = 0.72, or 72%
If the interface shows only the first number, the real chance of getting what you want is lower than the display suggests. Multi-step outcomes shrink quickly. Three independent 70% steps in a row succeed 0.7 × 0.7 × 0.7 = 0.343 of the time, about 34%. This is the multiplication principle at work, and it explains why a plan that depends on a chain of “pretty likely” events fails more often than it looks like it should.
Games differ in whether they show the combined figure or just one part, so it is worth checking what a displayed percentage actually refers to: hitting, damaging, killing or something else.
When the displayed number is not the true number
In some games, the percentage shown is deliberately not the underlying chance. One approach that has been used in some tactics games is to draw two random numbers and average them, instead of using one. Averaging makes results bunch toward the middle, which changes how the displayed value translates to real outcomes.
Illustration: suppose a game draws two numbers independently, each uniformly between 0 and 100, averages them, and counts a hit if the average falls below the displayed value.
- At a displayed 75%, the true chance works out to 1 − 2 × (0.25)², which is 1 − 0.125 = 0.875, or 87.5%.
- At a displayed 25%, the true chance is 2 × (0.25)² = 0.125, or 12.5%.
- At a displayed 50%, the true chance stays 50%.
The effect is that displayed chances above 50% are better than shown and displayed chances below 50% are worse. A designer might choose this so that likely events feel more reliable and unlikely ones feel rarer, matching what players seem to expect. This is only one example, and many games do not do it; where a game uses a method like this, it may or may not say so.
The same family of tools includes systems that adjust chances after streaks, described in the article on streak breakers. A game may combine several, which makes it hard to know the real chance from the interface alone.
Hidden rolls and timing
Some games decide the outcome before the player sees the result. A roll might be made when the action is confirmed, when the animation starts, or when the turn begins. Players who reload a save and try again can sometimes get a different result and sometimes the same one, depending on whether the random state is saved with the game. Whether a game reseeds or restores the generator at load time varies, and it is a common source of “the game remembers my failure” stories.
The relationship between a seed and a repeated outcome is explained in the article on pseudo-random generators.
Reading displayed percentages sensibly
- Treat a shown chance as an estimate of how often something happens over many attempts, not a prediction for the next one.
- Multiply chances together when several things must all go right.
- Allow for a handful of unlucky results in any long game, since they are likely to occur.
- If you suspect the numbers are off, track results over a large sample, such as a few hundred attempts. Even then, a small gap can be chance, and the more data you collect the more reliable the comparison becomes. Anecdotes from ten shots say very little.