Ludo Dice Odds: The Math Behind Every Roll

Ludo is played with a single six-sided die, so nearly every decision in the game rests on a handful of simple probabilities. How long will it take to get a token out? How dangerous is it to stop four squares in front of an opponent? Is it worth keeping a token back for the endgame? This guide works through the numbers so you can answer those questions with confidence.

Everything here assumes a fair die and the rules used on Thunder Ludo: a six to leave the yard, a bonus roll after every six, a forfeited roll on the third six in a row, and an exact roll to finish. The figures are exact unless we say they come from our simulation.

The basics of one die

Each face of a fair die comes up one time in six, about 16.7% of the time, and the average roll is 3.5. Just as important, a die has no memory: what happened on earlier rolls has no effect on the next one. Two sixes in a row will happen about once in every 36 pairs of rolls, and a long run without a six is not a sign that one is “due”.

Waiting for a six

At the start of the game, and whenever all your tokens are back in the yard, you need a six to move at all. With one roll per turn, the chance of having rolled at least one six grows like this:

The average wait is six turns, but the typical wait is shorter: half of all players get a six within four turns. The average is pulled up by unlucky streaks, and those are not rare. About one player in six is still waiting after ten turns. If that happens to you, it is normal variance, not bad technique.

This is also why the common house rule of three rolls per turn while all tokens are in the yard speeds up the opening so much. Three rolls give a 42.1% chance of a six every turn, which cuts the average wait from six turns to under two and a half.

How far a turn takes you

Because a six earns a bonus roll, an average turn covers more than 3.5 squares. Add the first roll (3.5 on average), the bonus roll that follows a six one time in six, and the third roll that follows two sixes one time in 36. That third roll only counts if it is a 1 to 5, since a third six is forfeited, so it is worth 2.5 on average:

3.5 + (1/6 × 3.5) + (1/36 × 2.5) = 299/72 ≈ 4.15 squares per turn.

That figure counts only the bonus rolls that sixes give you, not the extra rolls for captures or for bringing a token home, which push real turns a little further. It is a handy yardstick: a token 40 squares along has about ten turns of progress invested in it, which is exactly what you lose if it is captured.

How likely you are to be captured

Suppose a single opponent token sits some distance behind one of yours. What is the chance it can land on you during its next turn? The table assumes the worst case for you: the attacker always takes the hit if any of its rolls allow it, and can spend a six on a different token first if that helps.

Here is where the numbers come from. An attacker three squares behind you can hit you with a 3 (1 chance in 6), with a 6 spent elsewhere followed by a 3 (1 in 36), or with two sixes spent elsewhere followed by a 3 (1 in 216). Together that makes 43 chances in 216. If the attacker has no other token to spend a six on, the risk at one to five squares falls to exactly one in six.

The most useful line in the table is the drop between six and seven squares: from about 17 to 20% down to about 3%. Standing seven or more squares in front of an attacker is roughly six times safer than standing within six.

More than one attacker

With several opponent tokens behind you, the risks combine, but not always the way you might expect:

Two attackers at different short distances double the danger, because they need different rolls. Two attackers on the same square need the same roll, so the second adds nothing. Less obviously, an attacker exactly six squares further back than another adds nothing either: to reach you it needs a six followed by the very number the nearer attacker needs.

Why the last steps are slow

A token needs an exact roll to reach home. Whenever it is between one and six steps away, exactly one face of the die finishes it, so each roll has a one-in-six chance of taking it home. What changes is how often the roll can be used at all:

A lone token one step from home wastes five rolls out of six. Several tokens at different distances waste far fewer, because between them they can use more numbers. That is the mathematical reason behind the advice to keep your options open in the endgame.

How many turns a game takes

Each token must travel 56 steps from its start square to home, so four tokens need 224 squares of movement. Even if no roll were ever wasted, at 4.15 squares a turn that is about 54 turns, before counting the four sixes spent bringing tokens out.

Real players waste rolls: waiting for sixes, unable to use a big number near home. To get a realistic figure we simulated 100,000 solo races for each of three simple strategies, with no opponents and therefore no captures. Each simulated player used a fair die and the movement rules described above, and always finished a token when the roll allowed it.

Two things stand out. First, even a perfect race takes around 68 turns per player, and a real game takes longer, because every capture sends a token back to the start. Second, racing one token at a time is clearly slower, almost entirely because it leaves you with no legal move more than twice as often. Having more tokens in play gives every roll more places to go.

Three dice myths

“I haven’t rolled a six for ages, so one is coming.”

On a fair die, every roll is independent. After ten rolls without a six, the chance of a six on the next roll is still one in six.

“Sixes come in streaks.”

Streaks happen by chance and are memorable, which makes them feel more common than they are. On a fair die, two sixes in a row come up about once in every 36 pairs of rolls, and three in a row about once in 216 sets.

“Staying close behind an opponent is dangerous.”

It is the other way round. Danger runs forwards: the token behind is the threat, and the token in front is the target. Sitting one to six squares behind an opponent puts you in a strong attacking position, unless a different opponent is lined up behind you.

What it means at the table

• Expect to wait for sixes. Four turns is typical, ten or more happens to one player in six.
• Squares one to six in front of an opponent are dangerous; seven or more are far safer.
• Count attackers carefully. Two at different short distances double your risk.
• A token gains about four squares per turn, so a capture can cost ten turns of progress.
• Keep several tokens moving, and keep them at different distances near home.

For how to turn these numbers into decisions, read our Ludo strategy guide.