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 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.