Maximum value rolled on a handful of dice

Suppose you roll 2D6. Instead of adding the results together, you take the maximum of both dice. What would be the corresponding probability distribution?

(We will note the result of such a roll as max(2*D6) – to make distinction with the commonly used notation 2D6 which means adding the values.)

It’s of course intuitive to see that the probability of getting a result of 6 using such a roll would be higher compared to simply rolling a single D6. After all, there’s a higher probability of one of the dice scoring a 6. Likewise, the probability of rolling a 1 would be smaller, since this would require both dice rolling a 1, and this is only a 1/36 event compared to a 1/6 event when rolling a single die.

But what are the probabilities exactly?

As you can often do when computing probabilities for a low number of total outcomes, we can simply list all possible outcomes, and add up the probabilities for all relevant outcomes we wish to consider. The table below lists all possible combinations of rolling two ordinary D6, with the maximum value of both dice listed. We have highlighted all results that produce a maximum rolled value of 4.

max(2*D6)	1	2	3	4	5	6
1	1	2	3	4	5	6
2	2	2	3	4	5	6
3	3	3	3	4	5	6
4	4	4	4	4	5	6
5	5	5	5	5	5	6
6	6	6	6	6	6	6

Each cell in the table has a probability of 1/36, so if we add everything together, we have the following overall probability table for max(2*D6):

Result of max(2*D6)	probability
1	1/36 = 2.78%
2	3/36 = 8.33%
3	5/36 = 13.89%
4	7/36 = 19.44%
5	9/36 = 25%
6	11/36 = 30.56%

A first thing you can see is that the probabilities are not evenly distributed, as expected. On closer inspection, the probabilities go up in a linear trend: each results shows an increase of 2/36 = 5.56% over the previous one.

This also becomes obvious when we plot the values in a graph (all graphs in this blogpost are made using the online tool anydice.com).

A similar reasoning can be made for plotting the probability outcomes of the minimum of 2 dice. The graph below shows both max(2*D6) and min(2*D6).

And of course, varying the die type doesn’t make much of a difference for our analysis. The graph below shows plots for the maximum of 2 dice, for the usual type of dice used in games.

What about the maximum of rolling 5 D10’s?

So what if we roll more dice? Again, when rolling more and more D6’s, you would expect higher and higher probabilities of rolling a 6 as the maximum value. To compute these probablities exactly, let’s look at the table more carefully.

To count all cells with a result equal to 4, we can also count all cells with a result equal to or less than 4 (green and orange cells), and subtract the cells with a result equal to or less than 3 (green cells only). This leaves us with the number of cells that have a value exactly equal to 4 (orange cells).

Max(2*D6)	1	2	3	4	5	6
1	1	2	3	4	5	6
2	2	2	3	4	5	6
3	3	3	3	4	5	6
4	4	4	4	4	5	6
5	5	5	5	5	5	6
6	6	6	6	6	6	6

 As you can see, the green cells (result equal to or less than 3) form a  square, and the number of green cells can be expressed as the square of 3 or 3*3 = 9. Likewise, the number of cells with a result equal to or less than 4 add up to 4*4 = 16. Thus, the number orange cells (result equal to 4) equals 16 – 9 = 7. Dividing 7 by 36 (the total number of possible outcomes) gives us the desired probability 7/36.

In general, we can compute the probability of max(2*D6) being equal to x as:

Prob(x) = (x*x – (x-1)*(x-1)) / 36

We can generalize this to a higher number of dice. If we would roll 3 dice, the above square becomes a “cube” in the (3-dimensional) table, and we count the number of cells as x*x*x. For n dice, we go to an n-dimensional space, and we can write:

Prob(x) = (xⁿ – (x-1)ⁿ)/6ⁿ

If we want to roll a Dy with y faces instead of an ordinary D6, the formula becomes:

Prob(x) = (xⁿ – (x-1)ⁿ)/yⁿ

The graphs below show the resulting probabilities for rolling an increasing number of D10, and each time taking the maximum value as the result.

 The probabilities of rolling higher numbers go upward as you roll a higher number of dice, as can be expected.

Cumulative probabilities

In a recently published fantasy ruleset (Oathmark, Osprey Games), there is a mechanism in which you roll max(2*D10), and you need to roll equal or higher than a target number to activate a unit. Thus, we are interested in the cumulative probability: what is the probability of rolling *at least* a given value.

For this, we need to sum all probabilities for that target number or higher. The resulting graphs, again for an increasing number of D10, are shown below.

One can wonder whether this specific mechanic is worth it. Does rolling max(2*D10) really make a difference compared to some other – mechanically more familiar or simple die rolls?

The graph belows shows the cumulative plot (rolling equal or higher than a target number …) for max(2*D10), tha familiar 2D6, and D8+3.

As you can see, the differences are negligible, especially if you look in the range 5 to 10, which is the interesting range for activation target numbers.

Conclusion

Personally, I don’t really like the highest-roll-mechanic. It requires rolling multiple dice (and using only 1 as the final result), and it can easily be replaced by a more familiar die roll.