3 Dice Probability Chart (With Probabilities)


When you roll 1 dice, there are 6 possible numbers the dice could land on: 1, 2, 3, 4 5, or 6.

When you roll 2 dice, there are 6 * 6 = 36 possible combinations of numbers the dice could land on.

When you roll 3 dice, there are 6 * 6 * 6 = 216 possible combinations of numbers the dice could land on.

For example:

  • The first dice may land on 1,the second may land on 1 and the third may land on 1.
  • The first dice may land on 1,the second may land on 1 and the third may land on 2.
  • The first dice may land on 1,the second may land on 1 and the third may land on 3.
  • . . .

And so on.

It turns out that there is only 1 way for the sum of the dice to be 3:

  • First Dice = 1, Second Dice = 1, Third Dice = 1

However, there are 3 ways for the sum of the dice to be 4:

  • First Dice = 1, Second Dice = 1, Third Dice = 2
  • First Dice = 1, Second Dice = 2, Third Dice = 1
  • First Dice = 2, Second Dice = 1, Third Dice = 1

And there are 6 ways for the sum of the dice to be 5:

  • First Dice = 1, Second Dice = 1, Third Dice = 3
  • First Dice = 1, Second Dice = 2, Third Dice = 2
  • First Dice = 1, Second Dice = 3, Third Dice = 1
  • First Dice = 2, Second Dice = 1, Third Dice = 2
  • First Dice = 2, Second Dice = 2, Third Dice = 1
  • First Dice = 3, Second Dice = 1, Third Dice = 1

We can create the following chart to visualize the probability that the sum of the three dice is equal to a particular number:

3 dice probability chart

We can see that the probability distribution is symmetrical.

The most likely sum of the three dice is 10 or 11 while the least likely sum of the three dice is 3 or 18.

Additional Resources

The following tutorials explain other common topics in probability:

How to Find Probability of At Least One Head in Coin Flips
How to Find Probability of Rolling Doubles with Dice
How to Find the Probability of Neither A Nor B
How to Find the Probability of A or B

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