King of Tokyo Dice Odds: What to Keep After Your First Roll

Welcome to Level 2. At Level 1 you learned to pick a goal before you roll and to buy cards that keep paying. Now you put numbers on the dice. King of Tokyo is a dice game, but every die is fair and every face is known, so the odds can be worked out exactly. You do not need to memorize them. A few landmarks will change what you keep.

All the numbers here are our own calculations for the base game: 6 black dice, three rolls, and each die showing each of its six faces one time in six.

The one number to remember

If you chase a single face and reroll every die that misses, each die has about a 42 percent chance of showing that face by the end of your third roll. That is one minus the chance of missing three times in a row: five in six, cubed.

Everything else follows from that. With six dice chasing one face, you average about 2.5 of it, and here is how often you reach each count:

That table works for Smash, Energy, Heal, or a single number. The endgame lesson uses the same table to count a knockout.

Keep the pair, or let it go?

The most common decision in the game comes after the first roll: you see two of a number and wonder whether to chase the set.

• A pair, four dice and two rolls left: about 77 percent to get the third one.
• A single, five dice and two rolls left: about 48 percent to get two more.
• Nothing yet, starting fresh with six dice: 50 percent to reach three of a face, as the table shows.

So a pair of 3s on the first roll is a strong start. Keep it, and three times in four you score at least 3 points. But a pair of 1s on the same odds is worth only 1 point, and the four other dice you spend chasing it could have bought Energy or Smash. A pair is a reason to chase only when the set is worth more than what else those dice would do.

What each goal pays, on average

Here is what a whole turn is worth when you go all in on one goal from the first roll, keeping only the face you chase:

Two things stand out. First, chasing 1s from scratch is a weak turn: under a point on average. Second, a turn chasing 3s pays less than the 2 points you get for free by starting a turn in Tokyo. Numbers are a fine way to score, but Tokyo is the steadiest source of points in the game.

Of course you rarely go all in on one goal. These averages are a yardstick. When a turn could go two ways, compare roughly what each would pay.

Two goals, not three

Splitting your dice cuts every number above. Keeping 3s and Energy at once is fine, because a die that misses one goal can still land on the other. Chasing 3s, Energy, and Smash together usually ends with a little of each and none of what you needed.

A useful rule: name your main goal and one backup. Keep dice that serve either. Reroll the rest.

Remember the third roll

On your third roll, you may pick up dice you set aside earlier. Use it. If your first two rolls left you with two Energy and two 1s, and the turn has turned into a Smash turn because the Monster in Tokyo is low, the 1s can go back in the cup.

What an extra die or reroll is worth

Some Power cards add a die or a roll. Here is how they change the one-face chase:

An extra die raises your average by about a sixth. An extra roll does even more for a single goal, because every die gets another chance. Either way, the gain applies to every turn for the rest of the game, which is why the Power cards lesson put better dice first.

And the reverse: a Shrink Ray token takes a die away. With 5 dice, your chance of three of a face drops to about 35 percent.

The habit

After the first roll, ask two questions. What is this turn for? And which dice serve it? Then let the odds settle anything close. Most turns are not close at all, once you see the numbers.