 # Dice Odds for Every Type (d4, d6, d8, d10, d12, d20)

Unless you're playing Craps in Vegas, it's usually not necessary to calculate odds every single time you roll some dice. But it's good to have at least a general knowledge of dice odds, whether you're playing a tabletop role-playing game or a few rounds of Farkle. What is the chance of rolling a 6 on an 8-sided die? How about the chance of rolling a 10 on two 6-sided dice? Or what are the odds you'll roll a Yahtzee in a single roll of five dice?

Without getting into heavy-duty statistics, let's start by taking a look at the odds of rolling any single number on each die type. In other words, what are the chances of rolling that 6 on the 8-sided die, or rolling exactly a 1 on a 20-sided die?

Odds of rolling one specific number on each type of polyhedral dice:

d4 = 25%

d6 = 16.7%

d8 = 12.5%

d10 = 10%

d12 = 8%

d20 = 5%

What about rolling a certain number or greater? For example, what are the chances of rolling a 5 or higher on a 6-sided die, or a 12 or higher on a 20-sided die?

This table shows the percentage chance of rolling a particular number or higher for each type of polyhedral die.

 1 2 3 4 5 6 7 8 9 10 11 12 d4 100% 75% 50% 25% d6 100% 83% 66% 50% 33% 17% d8 100% 88% 75% 63% 50% 38% 25% 13% d10 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% d12 100% 92% 83% 75% 67% 58% 50% 42% 33% 25% 17% 8% d20 100% 95% 90% 85% 80% 75% 70% 65% 60% 55% 50% 45%

To save space this table only goes to 12, so for a d20 just reduce the percentage chance by 5% for each number beyond 12 (e.g., 13 = 40%, 14 = 35%, etc.)

What are some of the odds if you're rolling two 6-sided dice? In the table below, you can see that there are 12 possible numbers, of course, but each number has one or more possible combinations to roll that number. For example, there's only one possible combination for rolling a 2 on two d6s: 1 and 1. However there are 6 different possible combinations for rolling a 7: 6 and 1, 5 and 2, 4 and 3, 3 and 4, 2 and 5, and 1 and 6.

 Value Possible Combinations Chance of rolling an exact number 2 1 2.8% 3 2 5.6% 4 3 8.3% 5 4 11.1% 6 5 13.9% 7 6 16.7% 8 5 13.9% 9 4 11.1% 10 3 8.3% 11 2 5.6% 12 1 2.8%

And here are the chances of rolling above or below a certain number with two 6-sided dice:

 This number or higher This number or lower Chance to roll 2+ 12- 100% 3+ 11- 97.2% 4+ 10- 91.7% 5+ 9- 83.3% 6+ 8- 72.2% 7+ 7- 58.3% 8+ 6- 41.7% 9+ 5- 27.8% 10+ 4- 16.7% 11+ 3- 8.3% 12+ 2- 2.8%

Now let's talk about the mean, or “average,” for each type of dice. Why is this important? Knowing the average of each type of dice is useful because it allows you to be able to calculate your own odds when necessary. But what exactly is the mean? It is the average of all the numbers that you could expect to roll on a single toss of a die. So for example, the mean of a 4-sided die is 2.5. Obviously the number 2.5 is not a number that you can actually roll on a d4 (well, not on any typical d4 anyway), but it represents the average of all the numbers that you could expect to roll on a single toss of a 4-sided die. Any time you roll a d4, you have an equal chance of rolling any of the four numbers, so to calculate the mean (average), we add up all the numbers (1+2+3+4 = 10) and then divide that sum by the total number of numbers on the die (in this case, 4).

(1+2+3+4) / 4 = 2.5

Let's look at a 6-sided die next. The mean for a d6 is 3.5, calculated like so:

(1+2+3+4+5+6) / 6 = 3.5

And here is the mean for all the different types of dice:

d4 = 2.5
d6 = 3.5
d8 = 4.5
d10 = 5.5
d12 = 6.5
d20 = 10.5

Now that we know the mean for all those dice types, we can figure out what your average roll will be when you add in modifiers such as +5 or -2. If you play D&D or other tabletop role-playing games, you'll be familiar with this. For example, in D&D if you're attacking a monster with a short sword and you're applying a +2 proficiency bonus and a +3 Strength bonus, what can you expect that your average roll will be on a 20-sided die? You can calculate this by simply taking the mean of a d20 and applying the bonuses. In this case it would be 10.5 (the mean of a d20) plus 5 for the bonuses for a total of 15.5.

10.5+2+3 = 15.5

You can use this method to calculate the average for saving throws and skill checks too. Of course, just because your average number goes up doesn't mean you're guaranteed a successful outcome. It's still possible to roll a 1 after all!

What about if you're rolling multiple dice of the same type? The same method applies, but you'll have to calculate the mean based on the number of dice you're rolling by taking the mean of the die type and multiplying it by the number of dice. So in the case of rolling three 6-sided dice, just multiply the mean of a d6 by the number of d6s:

3.5 x 3 = 10.5

The average number you could expect to roll on three d6s would be 10.5!

More advanced odds and statistics are beyond the scope of this article, but check out  Omnicalculator and AnyDice to get the statistical probabilities for just about any type of dice roll.

Sep 1st 2014 Dice Game Depot

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