The calculations that do most of the work
Most in-game decisions that benefit from numbers rely on only a handful of calculations. None needs more than basic arithmetic. This article lays them out with worked examples so you can check your own reasoning, in review or in the moment.
Percentages, fractions and odds
A probability is a number between 0 and 1, or equivalently 0 to 100 percent. The same chance can be written several ways:
- 0.25, 25 percent, or one in four.
- 0.5, 50 percent, or one in two.
- 0.1, 10 percent, or one in ten.
Odds are a slightly different way of writing it. Odds compare the chances for and against. A 25 percent chance means one success for every three failures, so the odds are 1 to 3 (or “3 to 1 against”). A 50 percent chance is 1 to 1. Mixing up probability and odds is a common source of error: 1 to 3 is a probability of 1 in 4, not 1 in 3.
The complement
The chance that something does not happen is one minus the chance that it does. If a dodge works 30 percent of the time, it fails 70 percent of the time. This looks trivial, but it is the key to several other calculations.
Both things happening
When two independent things must both happen, multiply their chances. If you need a 50 percent skill shot and, separately, a 50 percent enemy mistake, the chance of both is 0.5 × 0.5 = 0.25, or 25 percent. With three such events it falls to 0.5 × 0.5 × 0.5 = 0.125, or 12.5 percent.
The practical lesson is that plans needing several things to go right are usually less likely than they feel. Each extra requirement is a multiplier below one.
The word “independent” matters. If one event makes the other more likely, such as a stunned enemy being easier to hit, simple multiplication will give the wrong answer.
At least one success
For the chance that at least one of several tries works, it is easiest to compute the chance that none work and subtract from one. Say you have two independent attempts, each with a 30 percent chance. Each fails 70 percent of the time. Both fail 0.7 × 0.7 = 0.49. So at least one works with chance 1 − 0.49 = 0.51, or 51 percent.
Notice that two 30 percent tries do not give 60 percent. They give 51, because the second try is only useful in the cases where the first one failed. The same reasoning underlies expected attempts for rare events: the average number of tries is a guide, while any individual run can be much shorter or longer.
Expected outcomes
An expected outcome weights each possible result by its chance. Suppose a play gives +3 points with 40 percent chance and −1 point with 60 percent chance. The average is 0.4 × 3 + 0.6 × (−1) = 1.2 − 0.6 = 0.6 points per attempt.
This does not mean you get 0.6 points on any single try. You will get +3 or −1. The figure describes the average over many repeats, which is why a single result says little about whether the play was sound. The same idea applied to a late-match choice is covered in the discussion of expected value, where the thing being weighted is usually your chance of winning.
Break-even chance
A particularly useful calculation is the success chance at which a risk stops being worth it. If a play costs you 1 unit when it fails and gains you 3 when it works, let p be the chance of success. The average value is 3p − 1(1 − p) = 4p − 1. That is above zero when p is above 1/4, or 25 percent.
A quick check: at p = 0.25, the value is 0.25 × 3 − 0.75 × 1 = 0.75 − 0.75 = 0. Good. At 40 percent it is 0.4 × 3 − 0.6 × 1 = 1.2 − 0.6 = 0.6, matching the previous example.
The general rule: if you risk R to gain G, break-even is R divided by (R + G). With R = 1 and G = 3 that is 1/4. With R = 2 and G = 2 it is 1/2. The bigger the payoff relative to the risk, the lower the success rate you need.
Rates versus counts
Sometimes you have data, not a chance. If a play worked 6 times out of 10 attempts, a natural estimate is 60 percent. But a small sample size is a rough guide: ten tries could easily come from a true rate somewhere well above or below 60. Treat early estimates as provisional, and expect them to move as you get more attempts. Careful stat tracking over a longer run, ideally in a simple spreadsheet, makes the estimates more reliable, though they will always carry some uncertainty.
Common slips to avoid
- Adding instead of multiplying. Two 30 percent chances are not 60 percent.
- Confusing odds and probability. 1 to 3 odds is 25 percent.
- Using the average for a one-off. The average describes many repeats, not the single attempt in front of you.
- Treating estimates as exact. A result like 0.6 comes from guessed inputs; if the inputs shift, so does the answer.
- Ignoring dependence. If events affect each other, multiplication needs adjusting.
Using the calculations
These tools are most useful in review, where you have time to check the numbers, and as background intuition in a match. In the moment, the rough versions are enough: halve, quarter, multiply the fractions, and ask whether the payoff justifies the risk. The point is not to remove uncertainty, which cannot be done, but to keep your reasoning honest about it.