Carcassonne Odds: Will the Tile You Need Come?

Welcome to Level 3. At Level 2 you learned to count how many tiles fit a space. Now you put that count to work. Carcassonne is a game of hidden draws, but the draws come from a known set, and that means you can put real odds on the question every strong player asks: will this close?

You do not need a calculator. You need three numbers and a feel for one table.

The three numbers

• Fits left: how many tiles still in the stack can fill the space. The tile counting lesson shows how to find it.
• Tiles left: how many tiles remain in the stacks.
• Your draws: how many of those you will draw. In a two-player game, about half of the tiles left. In a four-player game, about a quarter.

The next tile

The chance that the very next tile you draw fits is simply fits left divided by tiles left.

At the start of the game, 22 of the 71 tiles can close a one-sided city when the space beyond it is open, so a fresh draw closes it about 31% of the time. Later, with 10 fits among 30 tiles left, it is about 33%.

That is the easy part. The more useful question is about your next several draws.

Your next few draws

The chance you draw at least one fitting tile in your next few turns is one minus the chance that you miss every time. The arithmetic is simple but tedious, so here is the table worked out for 30 tiles left in the stacks.

With 30 tiles left, a two-player game gives you 15 more draws, and a four-player game about 7 or 8.

Two things stand out:

• One or two fits left is a long shot unless you have many draws to come.
• Draws matter as much as fits. Three fits with 15 draws is a near certainty. Three fits with 5 draws is a coin flip, slightly against you.

A quick estimate when you have no table: multiply fits left by your draws and divide by tiles left. That is roughly how many fitting tiles you expect to draw. When the answer is well under 1, the chance of drawing one is a little below that number.

Two cautions

Other players draw too. The table is the chance that you draw a fit. A rival who draws one might place it somewhere harmless, or might use it to fill your space with a shape that closes nothing. On a shared feature, a rival may even close it for you.

Spaces change. Every tile placed next to your space can add a new side to match. The number of fits can drop from 22 to 5 in one turn. Recount when the neighbors change.

Putting odds to work

Odds become decisions when you compare values. Take a four-player game with 30 tiles left. Your city has 6 tiles and 1 coat of arms, so it is worth 14 completed and 7 unfinished. Its last open space is boxed in by fields, so only plain city-edge tiles fit. Two are already on the table, so 3 are left. You will draw about 7 more tiles.

From the table, 3 fits and about 7 draws is roughly a 55% chance to close. The city’s expected value is about 7 plus 55% of the other 7, or close to 11 points.

Now ask:

• Should you add a second knight to defend it? Only if the extra meeple would not do better elsewhere for the rest of the game.
• Should you make it bigger? Adding tiles adds points, but if a new tile opens another side, you multiply the risk. One more open side with few fits means you now need two rare tiles, not one, and your odds drop sharply.
• Should you give up on it? At close to coin-flip odds, no, but stop adding to it and spend your next meeples on things that finish.

The habit

You will not run these numbers every turn, and you should not. Run them for the two or three biggest features on the board, especially your own city and the one you are thinking of joining. After a few games the table becomes instinct: you will feel the difference between three fits and thirty tiles and three fits and ten.