The 3 in Catan
2 of the 36 ways two dice can land make 3, so you see one about every 18.0 rolls — 5.6 times in 100. Among the ten production numbers that makes it the 4th least productive number of 10 — tied with the 11, which is rolled exactly as often.
Why the pips are the odds
The dots on a number token are not decoration and they are not a rating someone assigned. They are a count: 2 pips because there are exactly 2 ways for two dice to make 3. Read the pips and you have read the probability.
- 1 + 2
- 2 + 1
Ways to roll each sum with two dice. 7 is the peak, which is why it moves the robber instead of producing anything.
What a settlement here is worth
A settlement on a single 3 hex earns 0.056 cards per roll — 5.6 over 100 rolls, and double that for a city. Against the weakest numbers on the board that is 2× the production, which is the entire reason placement matters more than almost any later decision.
One caveat worth keeping: this is one hex in isolation. A real spot touches up to three, and if two of them share a number they fire together — one correlated event, not two independent ones. The resource odds calculator handles that properly for a board you actually build.