Dice That Ace More Do NOT Roll Higher
Every few months, I come across someone online or in real life who holds an opinion like this for Savage Worlds (although it could also apply to Cortex):
Lower dice are better because they ace more! I mean, a d4 has a 1 in 4 chance of rolling a 4, meaning that you get to roll it again and add 4. But a d6 only has a 1 in 6 chance of rolling it again and adding 6. In fact, you get diminishing returns the higher you go up.
I’d like to dispel once and for all that this belief is wrong! While the probabilities of acing (also called “exploding”) more are indeed higher on lower dice, the truth of the fact is that, if you calculate the odds, you still have a better chance of rolling higher on higher dice, despite the fact that they ace less. And although there are certain TNs that are easier to reach on lower dice in some instance, when you consider that you get the same result when you consider the “Raise Bracket” (that is, the TN and the 3 TNs above it, all of which result in the same thing), there is no difference.
Average Die Rolls for Normal Dice
First, let’s start off by calculating the average die rolls for normal, non-acing dice. We can do this by simply adding up all the numbers on each of the sides, then dividing by the number of sides. The average die roll for normal dice is as follows:
- d4 – 2.5
- d6 – 3.5
- d8 – 4.5
- d10 – 5.5
- d12 – 6.5
Some people, when seeing these numbers, are surprised to find that the averages are actually .5 higher than they expect. Many people take the mental shortcut of taking the number of sides and dividing by 2, thus “averaging” the high and low values and hopefully arriving at the middle. It’s close, but it’s not the right answer. I say all of this because I think it’s a simple instance of how our minds take mental shortcuts to figure out complex odds, such as what a die will roll.
Average Die Rolls for Normal Dice
Calculating the average die roll of an acing die is a bit more difficult because, as the definition says, it is open ended. But there are lesser and lesser odds of getting higher and higher numbers, to the point where the chances of getting an extraordinarily high number are so miniscule that it doesn’t have any meaningful bearing on the average die rolls.
This article provides all the math to show how to get the value of an exploding die. With that in mind, we discover that the odds of exploding dice are as follows:
- d4 – 4.17
- d6 – 4.9 (.73 higher)
- d8 – 5.78 (.88 higher)
- d10 – 6.11 (.97 higher)
- d12 – 7.09 (.98 higher)
Note that even though higher dice ace less, the average value of each die is still higher than the value of the die below it and the rate of change increases the higher you go. So rather than having diminishing returns the higher you go because you ace less, you have increased returns because the number of sides increases despite acing less.
I’ll add the disclaimer that if you are trying to reach certain target numbers, there are very rare instances where a die one step lower has about a 1% greater chance of reaching that target than the higher die (I tried to disprove that, but ultimately wound up finding that such rare instances did exist). Still, it’s a 1% chance in rare instances, and overall, higher dice are still better because, on average, they roll higher.