How the calculator works
Two fair six-sided dice have 36 equally likely ordered outcomes. A 6 appears in five of them, so one settlement beside a 6 ore tile earns 5/36 of an ore card per roll on average. A 5 grain tile adds 4/36 of a grain card. We add the contributions from every producing tile beside your two settlements.
Expected cards per 36 rolls = the sum of the dice weights for all producing slots. These weights are 1, 2, 3, 4, 5, 5, 4, 3, 2, 1 for numbers 2, 3, 4, 5, 6, 8, 9, 10, 11, 12. For the chance of receiving anything, each distinct number counts only once. Two tiles on 6 can pay two cards together, but the chance of rolling 6 is still 5/36.
Use the result to ask a better question
The first example compares 21 expected cards spread across all five resources with 23 concentrated in ore, grain and wool. The second opening has higher raw output, but no direct road materials. Decide how it will expand before choosing it. Neither total includes the price of the trades you might need.
The second example keeps expected output at 23 while moving brick from 6 to 8. Separate numbers give more opportunities to collect something; shared numbers can deliver wood and brick together. This is a difference in the pattern of production, not evidence that a number becomes due after a drought.
What this model leaves out
Assume an unblocked board, enough bank supply and one settlement beside each selected slot. The model excludes the robber, discards, trades, ports, starting cards, cities, development cards, opponent decisions and future expansion. It does not validate full-board geometry or rank legal placements. Coast and desert contribute zero. If both settlements touch the same tile, include that tile once for each settlement: it pays each of them.
The fair-dice formula applies to the base tabletop game and random-dice settings. Colonist’s balanced modes and Archipelago’s dice cycle are different processes. Do not use these figures as their conditional next-roll odds. Read Archipelago’s dice explanation.
Sources and next reading
The probability model is an exact enumeration of two fair dice. CATAN’s official game page describes its production and number probabilities; Colonist documents its balanced-dice approach separately. The worked comparisons and interpretation here are original editorial examples, not measured win rates.