Theory & Strategy: Average Dice

The number one way to improve your results in dice-based games is to understand how dice work. In particular, it’s important to understand the concept of average dice. Used correctly, it will bend results in your favor. Understood incorrectly, it will trick you into sub-optimal play and frustrating, seemingly incomprehensible defeats.

When people say they will succeed “on average dice” they mean that the most common result, and any result that’s better, will be enough. Without getting into all the math (there’s a good primer here), 7 is the most common result when rolling two dice. Thus, if all you need is a 7, you will usually succeed. The most common result, and all the numbers above the most common result, will work for you.

Once you know what the average result is, you can evaluate your options in dice-driven games much more effectively. If you need an above-average result to succeed, you will probably fail. You can’t rely on something working if you need an 8, and if you need an 11 it’s a true long shot. By contrast, if a less-than-average total is enough the odds are strongly in your favor. They become all the more so as the required total gets lower; it’s easy to roll 5 or more, and you’ll get a 3 or more almost every time.

I play a lot of dice-based wargames, and the difference an understanding of average dice makes in people’s win record is astonishing. They can objectively determine what is likely to work, rather than being seduced by the promise of what might work. Nothing leads to winning like consistently getting positive results from each move, and applying the concept of average dice generates those results.

With all of that said, it’s important to remember that average dice are not guaranteed dice. An average roll is called “average” because half of the remaining possible results are lower. It is likely that you will succeed when all you need is the average, but there is still a substantial chance of failure.

This is especially important to remember when your plan involves multiple rolls. I often hear people say things like “I only need six average rolls.” The odds of rolling the most common result or better six times straight are not very good! It is much more likely that some of those rolls will fall short. Your strategy needs to be able to hang together when that happens.

Misunderstanding the likelihood of an average roll is especially devastating when high results can’t make up for low ones. For example, take to-hit rolls. In most games, whether the player hits the target is binary: either the roll was enough or it wasn’t. An excellent first roll can’t make up for a bad second roll; the first is a hit and the second is a miss, no matter how high the first one was. The excess from the first roll can’t be applied to make up the amount the second is lacking.

When high rolls can make up for low ones, things might even out such that needing “six average rolls” is less of a problem. (Even then you’ve got a good chance of ending up below average; it’s still a problem, just less of one.) When they can’t, however, relying on six average rolls in a row is a critical mistake. Hitting six times in a row, when you need average rolls each time, isn’t an average result. It’s a very unusual one, and you would need to be very lucky to pull it off.

Dice don’t hate you, but math doesn’t pity you, either. Strategies that demand better-than-usual rolls, or even multiple average rolls in sequence, generally don’t work out. Minimizing the number of rolls you have to make, and the results you have to get, will maximize your chance of winning.

2 thoughts on “Theory & Strategy: Average Dice

  1. Well said.

    One clarification:
    The average on 2d6 actually is exactly 7, in addition to being the mode. The mode and average match when combining a pair of any standard dice.

    And a tidbit no one ever asked:
    There’s a fairly easy shortcut to know the average of any [standard] die result: Half the max +0.5. A d4’s average is 2.5 and a d20’s is 10.5. When adding die results, you can add their individual averages to get the combined average, so 1d4+1d20 has an average of 13. (But the mode on that is weird because there are so many results tied for “most likely” due to the spread flattening of combining such disparate dice.)

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