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:
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
