Dice Probabilities, Part 2: Pools, Matches and Thresholds

Once you roll a pool of colour dice, most interesting questions are about groups rather than single faces. Three of the most common: do any colours match, do at least k dice show one colour, and are all six colours represented?

All different. With n dice and six colours, the chance that every die shows a distinct colour is 6!/(6−n)! ÷ 6^n for n ≤ 6. For three dice that is 120/216 ≈ 55.6%; for four dice 360/1296 ≈ 27.8%; for six dice only 720/46656 ≈ 1.5%. Beyond six dice a match is guaranteed by the pigeonhole principle.

At least k of a colour. Compute it as one minus the tail: P(at least k reds in n dice) = 1 − Σ from i=0 to k−1 of C(n,i)(1/6)^i(5/6)^(n−i). With 10 dice, at least three reds is about 22.4%; with 20 dice it is about 67.7%. If you want a threshold to trigger around half the time, size the pool so k sits just above n/6.

All six colours present — the coupon collector problem. You need on average 6 × (1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6) = 14.7 dice to see every colour once. With exactly 12 dice the probability of a full set is about 43.8%; with 20 dice about 78%; with 30 dice about 94%.

Designing a game around these numbers is mostly about choosing where the interesting uncertainty sits. Outcomes below roughly 10% feel like jackpots and should pay accordingly. Outcomes above 80% stop being decisions and become taxes. The band between 25% and 65% is where players actually feel they are gambling.

One practical warning: simulating a rule is faster than deriving it. Roll a hundred dice a few dozen times, tally the outcome you care about, and you will get a usable estimate long before you finish the binomial sum by hand.