Probability You Actually Need
Core probability tools for tuning dice, decks, and draw odds in game design.
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Questions Covered in This Set
11 cards to master
Basic probability formula for fair dice and shuffled decks
favorable outcomes / total outcomes — probability is just counting.
How many ordered outcomes when rolling 2d6, and what is P(7)?
36 outcomes; 7 occurs 6 ways, so P(7) = 6/36 ≈ 16.7% (the peak of the curve).
Design lever: adding dice vs. adding faces
Adding dice creates a bell curve (reliable center); adding faces creates a flat plateau (wild, anything-can-happen).
AND vs. OR rules for combining probabilities
Independent events AND → multiply (1/6 × 1/6 = 1/36). Mutually exclusive events OR → add (1/6 + 1/6 = 1/3). Don't add overlapping events.
The complement trick for 'at least one'
P(at least one success in n tries) = 1 − (1 − p)^n. E.g. P(at least one 6 on 2d6) = 1 − (5/6)² ≈ 30.6%.
How many d20 rolls before a natural 20 is more likely than not?
14 rolls: 1 − (0.95)^n > 0.5 at n = 14.
Expected value (EV) — definition and d6 examples
Long-run average: multiply each outcome by its probability and add. One d6 = 3.5; 2d6 = 7; a d6 attack that misses 1/3 of the time = 2/3 × 3.5 ≈ 2.33.
Why isn't EV enough for balancing?
EV lies about small samples — two abilities with EV 3 feel totally different if one is 0-or-9. Always ask about the spread (variance) too.
Why are decks different from dice?
Dice have no memory; decks are drawn without replacement (hypergeometric). P(ace) = 4/52, then 3/51 — multiply sequential fractions.
Chance of at least one resource in a 7-card hand from a 40-card deck with 17 resources
P(none) = (23/40)(22/39)...(17/34) ≈ 1.7%, so P(at least one) ≈ 98.3%.
How do you tune 'a 50% chance the event happens within 5 turns'?
Solve (1 − p)^5 = 0.5 → p ≈ 13% chance per turn.
