Why Two Rolls Can Reverse the Dice Advantage

Imagine choosing a die that beats your opponent’s more often than not. Now imagine keeping those same dice, rolling each one twice, and comparing the totals—and discovering that your opponent has the advantage instead.

Nothing about the dice has changed. They are still fair, and their average rolls are still identical. Only the scoring rule has changed.

That small change reveals something important: “wins more often” and “scores more on average” are different measures of success. Neither gives you a universal ranking.

Three Dice With No Best Choice

Consider these three fair, six-sided dice:

A fair die is one whose six faces are equally likely to land face up. Fairness does not require the usual labels 1 through 6.

Because every label here appears twice, you can simplify the arithmetic: each die effectively chooses among three values, each with probability one-third.

If you and an opponent roll once each, with the higher number winning, the results form a loop:

• A beats B with probability $5/9$.
• B beats C with probability $5/9$.
• C beats A with probability $5/9$.

That is about 55.6% in each case.

These are called nontransitive dice. A relationship is transitive when it carries through a chain: if A ranks above B and B ranks above C, then A ranks above C. These dice break that pattern, much like rock–paper–scissors.

There is no strongest die. There is only a stronger choice against a particular opponent.

Where the 55.6% Comes From

To see why A beats B, compare their three possible values:

There are nine equally likely combinations, and A wins five of them.

On the physical six-sided dice, there are 36 equally likely face combinations. Each cell in this smaller table represents four of those combinations, so the same probability is $20/36=5/9$.

The other two matchups work similarly:

• B’s 1 beats nothing on C, its 6 beats two values, and its 8 beats all three: five wins.
• C’s 3 beats one value on A, its 5 beats two, and its 7 beats two: again, five wins.

Equal Averages Do Not Mean Equal Chances of Winning

The expected value of a roll is its probability-weighted average—the average you would approach over many repeated rolls.

For A, that is:

B and C also have expected value 5.

But expected value and win probability ask different questions:

The key difference is that the average accounts for the size of a score, whereas a win counts the same whether you win by one point or eight.

For example, A’s 9 beats every possible B roll. But A’s 2 and 4 usually lose to B. Those large and small scores balance out to the same average as B’s values, while still producing a slight advantage in the number of contests won.

An average summarizes one distribution. A win probability describes how two distributions meet.

Here, a distribution simply means the possible results together with how likely each one is.

Change the Rule: Roll Twice and Add

Now keep the same dice but change the game:

1. Choose a die.
2. Roll it twice, independently.
3. Add your two results.
4. Compare your total with your opponent’s total.

For A versus B, the advantage reverses:

These are exact probabilities, rounded for display—not estimates from a simulation.

A’s precise two-roll win probability is:

In words: A wins 37 of the 81 equally likely combinations of two effective three-value rolls per player.

If you count individual faces on the physical dice instead, there are $6^4=1,296$ equally likely outcomes: two rolls for A and two for B. A wins 592 of them, giving the same probability.

The New Totals Are Not Equally Likely

The important change is not just that the numbers become larger. The distribution of scores changes.

A can make 6 in two ways—$2+4$ or $4+2$—but it can make 4 only one way: $2+2$. So 6 is twice as likely as 4.

Here are all the possible totals. The “count” columns show how many of the nine equally likely ordered roll pairs produce each total.

Notice that the two dice have no totals in common, so there are no ties in this matchup.

To count A’s wins, take each A total and count how many B roll pairs it beats:

That is the reversal, laid out without any simulation or approximation.

Why Adding Rolls Is Not Just “More of the Same”

It is tempting to think that if A usually wins one roll, giving it more rolls should reinforce its advantage. But adding scores does not preserve the original contest.

Winning Twice Is Different From Winning the Total

Suppose the two rounds go like this:

• A rolls 9; B rolls 8.
• A rolls 2; B rolls 6.

Each die wins one round. But B wins the total, 14 to 11.

The sum keeps track of winning margins. A narrow win can be outweighed by a larger loss.

That means you cannot determine the two-roll result merely by knowing that A wins a single round with probability $5/9$. You also need to know which values produce those wins and losses.

A’s Highest Face Stops Being an Automatic Win

In the one-roll game, A’s 9 is unbeatable.

In the two-roll game, rolling one 9 does not guarantee anything. Paired with a 2, it makes 11; paired with a 4, it makes 13. B can exceed those totals with combinations of its 6s and 8s.

A’s 18 still beats everything, but it requires two 9s and occurs only one-ninth of the time.

Meanwhile, both dice now have expected total 10. Their averages remain equal—but their new score distributions favor B in head-to-head comparisons.

How Rare Is the Equal-Average Setup?

You can also explore how special this construction is.

Start with the numbers 1 through 9. Divide them into three unordered groups of three, then duplicate each number to make three six-sided dice.

There are 280 such divisions. “Unordered” means that swapping the names of the three groups does not create a new arrangement.

Because the numbers 1 through 9 sum to 45, equal averages require each group to sum to 15. Only two divisions satisfy that condition:

Both produce a circular winning relationship. In the second arrangement, $\{1,5,9\}$ beats $\{3,4,8\}$, which beats $\{2,6,7\}$, which beats $\{1,5,9\}$—again with probability $5/9$ for each winning matchup.

That does not mean equal averages always imply nontransitivity. It means that within this particular, tightly defined search, both equal-average arrangements have that property.

The Lesson: “Better” Needs a Rule

These dice offer two distinct surprises:

• Changing the opponent changes the better choice.
• Changing the scoring rule can reverse the better choice.

There is no contradiction. Averages, win rates, and summed scores measure different aspects of the same outcomes.

So whenever you are offered a single ranking, it is worth asking: better against whom, and under what rules?

Sometimes the most interesting part of a system is exactly what that ranking leaves out.