Expected Successes in a Dice Pool

A pool works by rolling several dice at once and counting how many meet a target, rather than by adding them up. The average number of successes is the per-die chance multiplied by the number of dice, and that is the entire calculation.

The reason it is that simple is linearity of expectation again, and it is worth pausing on because the result is stronger than most people assume. Write the count of successes as a sum of one-or-zero indicators, one per die. The average of each indicator is just the chance that die succeeds. The average of the sum is the sum of the averages. This holds whether or not the dice are independent — it would still be true if the dice were glued together — which is why no distribution appears anywhere in the derivation.

What you have arrived at is the binomial mean, $\mu = np$, reached without ever writing down the binomial distribution. That distribution is still what governs the actual spread: the variance is $np(1-p)$, so a pool of six dice at a 40% target averages 2.4 successes with a standard deviation of about 1.2 — which means results from 1 to 4 are all thoroughly ordinary, and zero successes out of six happens about once in twenty-one throws.

Two things the average cannot do. It is almost never a possible result — a pool returns a whole number of successes and 2.4 is not one — and it says nothing about the chance of clearing a threshold. "How often do I get at least three successes?" is a real and common question, and it needs the full binomial distribution summed over the qualifying counts, not this figure. What the average is genuinely good for is comparing designs: whether more dice at a harder target beats fewer dice at an easier one is answered here exactly, and answered in one multiplication.

Two assumptions to keep visible. Every die must be the same size and use the same target, or the sum of separate expectations is needed instead of a single product; and the dice must be fair. Neither is guaranteed by the arithmetic.

And the caution that closes every dice page in this shard, which applies with particular force to pools, because a handful of dice thrown together feels like one event with one mood. It is not. It is a set of independent rolls that happen to land at the same moment. A pool that came up empty last time has exactly this expectation on the next throw — not more, not less — and no run of poor throws is ever repaid. The long-run average arrives by swamping the early results, never by correcting them.