If you roll three dice at the same time, what's the probability of rolling a sum of 3 or 4?

Calculating the odds of a specific dice roll begins with figuring out the total number of possible outcomes. Since each of the three dice has six sides, you find the total by multiplying 6 x 6 x 6, which gives you 216 unique combinations. This number forms the foundation (Review) for our probability calculation. Out of all these possibilities, we only care about the ones that add up to 3 or 4.

To get a sum of 3, there's only one possible combination: each die must land on 1 (1-1-1). To get a sum of 4, there are three different ways: 1-1-2, 1-2-1, and 2-1-1. Even though they use the same numbers, the different positions on the three distinct dice make them separate outcomes. Adding these together, we have one way to get a 3 and three ways to get a 4, giving us a total of four "favorable" outcomes.

This type of problem is the bedrock of probability theory, a field of math that gained prominence in the 17th century as mathematicians like Blaise Pascal analyzed gambling puzzles. The distribution of sums from three dice forms a bell curve; the extreme results like 3 and 18 are the rarest, while middle-of-the-road sums like 10 and 11 are the most common because they can be formed in many more ways. With 4 favorable outcomes out of 216 total, the chance is quite small.