Dominion Odds: Collisions, Key Cards, and Hitting $8

Welcome to Level 3. At Level 2 you learned to build: money, trashing, and engines. Now you put numbers on it. Dominion feels like a game of lucky shuffles, but your deck is a set of cards you chose. You know exactly what is in it. That means you can work out the odds of the hands it will deal you.

Every number in this lesson is our own calculation from the card counts, using standard counting for a well-shuffled deck. You do not need to redo the math. You need a feel for a few tables.

The one-card rule

The chance that a particular card is in your next 5-card hand is 5 divided by the size of your deck. With a draw card like Smithy in hand, you see 7 or 8 cards instead, so use that number instead of 5.

Two lessons hide in that table. First, a single copy of a key card, like your Witch, shows up in about a quarter to a third of your hands in the midgame. If it has to show up more, buy a second copy or thin the deck. Second, draw helps you find it, but less than you might think.

How often terminals collide

A collision is two or more terminals in the same hand with only one Action to play them. Here is the chance of a collision in a 5-card hand.

This is why money decks stop at one or two terminals early and add a third only once the deck is large. Two terminals in 20 cards collide 1 time in 20. Four terminals in 15 cards collide 2 times in 5, which means many turns where a card you paid for does nothing.

The table counts collisions in your starting 5 cards only. Draw cards make collisions more likely, because a Smithy can draw your other Smithy. Treat these numbers as the floor.

How often a deck hits $8

The opening lesson showed that your first two hands always add up to $7. So a starting hand never reaches $6, and any one starting hand reaches $5 only about 1 time in 12. Here is how fast that changes as you add Treasure, for a money deck with no Action cards.

With a Silver and Silver opening, the chance of at least one $6 hand on turns 3 and 4 is about 42%, and of at least one $5 hand, about 91%. That second number is why so many openings plan for a $5 card on the next shuffle.

What greening costs

Look at the last two rows of the $8 table. The same deck that hits $8 about 41% of the time drops to about 23% after it buys three Provinces. Each Province you buy makes your next Province harder to buy.

That is the real reason timing matters. A deck that starts greening too early stalls, and a deck that waits too long may be out-raced. The endgame lesson gave you the rules of thumb. This table shows the cost behind them.

Shuffle timing

One more pattern: a card bought just before you shuffle joins the deck almost at once. A card bought just after a shuffle waits a whole pass through the deck. Late in the game, that matters. If your deck is down to its last few cards, a card you buy now will be shuffled in on the next turn or two. If you just shuffled, it may not appear before the game ends. When you are deciding between a Gold and a Duchy, a glance at how much deck you have left is part of the decision.

Using odds without slowing the game

You will not run numbers every turn, and nobody wants you to. Use them at three moments: before buying another terminal, when deciding whether a key card needs a second copy, and when deciding to start greening. After a few games the tables become a feel, and the feel is usually right.