Three fair dice are rolled, and all outcomes are equally likely. What is the probability that the sum of all the numbers on the dice is at most 6?

To find the probability that the sum of all the numbers on the dice is at most 6, we need to count the number of favorable outcomes and divide it by the total number of possible outcomes.

Let's break down the problem step by step:

Step 1: Determine the possible combinations of numbers that give a sum of at most 6.

Since three dice are rolled, the minimum sum we can get is 3 (1+1+1) and the maximum sum is 18 (6+6+6). We need to find the combinations that yield sums from 3 to 6.

The following combinations satisfy this condition:
- 1 + 1 + 1 = 3
- 1 + 1 + 2 = 4
- 1 + 2 + 1 = 4
- 2 + 1 + 1 = 4
- 1 + 1 + 3 = 5
- 1 + 3 + 1 = 5
- 3 + 1 + 1 = 5
- 2 + 2 + 1 = 5
- 2 + 1 + 2 = 5
- 1 + 2 + 2 = 5
- 1 + 1 + 4 = 6
- 1 + 4 + 1 = 6
- 4 + 1 + 1 = 6
- 2 + 3 + 1 = 6
- 2 + 1 + 3 = 6
- 3 + 1 + 2 = 6
- 1 + 2 + 3 = 6
- 1 + 3 + 2 = 6

Step 2: Calculate the number of favorable outcomes.

There are 18 possible combinations that give a sum of at most 6.

Step 3: Determine the total number of possible outcomes.

Since each dice has 6 sides and there are three dice rolled, the total number of outcomes is 6^3 = 216.

Step 4: Calculate the probability.

Divide the number of favorable outcomes (18) by the total number of possible outcomes (216) to get the probability:

P(sum of numbers at most 6) = 18/216 = 1/12 ≈ 0.0833

So, the probability that the sum of all the numbers on the dice is at most 6 is approximately 0.0833, or 1/12.