Chains of steps
Many things in games happen in stages. You must pass a hit check, then a damage check, then a save. A crafting item must survive several upgrade attempts. You need a particular card on the first draw and another on the second. In each case the question is the same: what is the chance that every step goes the way you want?
The multiplication rule answers it. When a result requires step one and step two and step three, you multiply the probabilities of the steps. The word “and” is the clue.
The rule for independent steps
If events are independent, meaning one does not affect the others, then:
P(A and B) = P(A) × P(B)
and this extends to any number of steps. Three consecutive sixes on a fair die:
1/6 × 1/6 × 1/6 = 1/216, about 0.46 percent.
Why does multiplying work? Imagine all the equally likely outcomes of the first two steps laid out in a grid. If step one succeeds in a fraction a of cases, and step two succeeds in a fraction b of cases regardless, then both succeed in a × b of cases. With two dice, 6 of the 36 outcomes have a six on the first die, and of those, 1 in 6 also has a six on the second, which is 1 outcome out of 36.
Long chains shrink fast
Multiplication of numbers below 1 makes the result smaller than any individual step. This effect is easy to underestimate.
Suppose an upgrade has a 70 percent chance to succeed, and you need five in a row, with each attempt independent. The chance of all five is 0.7^5 = 0.16807, about 16.8 percent. Each step looks safe, yet the full chain succeeds only about one time in six.
Or consider a dungeon with 10 rooms where you survive each with probability 0.9. The chance of clearing all of them is 0.9^10, about 0.3487, or 34.9 percent. Compare that with the 90 percent for one room, and you can see why reliability matters much more in long sequences. A small improvement per step has a bigger effect over a long chain: at 95 percent per room, ten rooms give 0.95^10, about 59.9 percent.
A compact way to see this: four upgrades at 50 percent each succeed together with probability (1/2)^4 = 1/16 = 6.25 percent.
When steps are not independent
Often the second step depends on the first. For example, drawing cards from a deck without replacement. The more general rule uses a conditional probability for each later step:
P(A and B) = P(A) × P(B given A)
Three cards, all hearts, dealt from a full deck:
13/52 × 12/51 × 11/50 = 1,716/132,600, about 0.0129, or 1.29 percent.
Each factor uses the cards that remain after the earlier hearts have been removed. If you had treated each draw as independent, using 13/52 three times, you would get (1/4)^3 = 1/64, about 1.56 percent, which is too high. The difference is the effect of dependence.
Deciding whether your steps are independent is the key judgement. The distinction between independent events and dependent ones determines which version of the rule applies, and the usual test is whether learning the result of one step would change your estimate of another.
Mixing “and” with “or”
Real problems mix the two connectives. A character must hit and then either crit or roll high on damage. The “and” parts multiply, and the “or” parts add, provided the alternatives cannot happen together.
As a simple example, suppose you need a six on a die, and then either a heads or a tails on a coin. The coin part is certain (1/2 + 1/2 = 1), so the combined probability is 1/6. More realistically, say a hit has probability 0.8, and given a hit, damage is “high” with probability 0.25 and “very high” with probability 0.1, which are mutually exclusive. The chance of a hit with at least high damage is 0.8 × (0.25 + 0.1) = 0.8 × 0.35 = 0.28, or 28 percent. The numbers here are illustrative, not from any particular game.
Probability trees
A tree diagram turns the multiplication rule into a picture. Each branch carries a probability, and the probability of any complete path is the product along it. The probabilities of all complete paths should sum to 1. Trees are helpful when there are several stages and different branches lead to different later options. They are also the idea behind game trees, which use a similar structure to compare choices.
For two dice-based steps, such as a 5/6 chance to avoid a trap followed by a 3/4 chance to win a fight, the path “avoid and win” has probability 5/6 × 3/4 = 15/24 = 5/8, or 62.5 percent. The path “avoid and lose” has 5/6 × 1/4 = 5/24, and a path with the trap triggered has 1/6. Adding them: 15/24 + 5/24 + 4/24 = 24/24 = 1. The check works.
Where the rule is used in practice
The rule underlies much of the arithmetic of game design. A designer who wants a rare event to occur about once in a few hundred tries can chain a few moderate checks. A player can use the rule to see how likely a plan is to work when it needs several things to go right. It also feeds into expected damage calculations, where hit chance and damage chance are multiplied together and then with the damage amount.
Common errors
- Adding instead of multiplying. Chances of “and” combine by multiplication, not addition.
- Using original odds for dependent steps. Update the probabilities after each draw or event.
- Ignoring that the result is a long-run frequency. A 16.8 percent chain still succeeds on the first try for some players and never for others.
- Forgetting that the order of independent steps does not matter. The product is the same in any order, so you can multiply in whichever sequence is convenient.
If a plan needs many things to go right, the multiplication rule will usually tell you the plan is shakier than it feels. That is useful knowledge, but it describes odds, not guaranteed results.