One popular probability problem is to roll a die. A standard die has six sides with numbers 1, 2, 3, 4, 5 and 6. If the die is fair (and we will assume that all of them are), then each of these outcomes are equally likely. Since there are six possible outcomes, the probability of obtaining any side of the die is 1/6. Thus the probability of rolling a 1 is 1/6, the probability of rolling a 2 is 1/6 and so on for 3, 4, 5 and 6. But what happens if we add another die? What are the probabilities for rolling two dice?

### What Not to Do

To correctly determine the probability of an event we need to know how often the event occurs, then divide this by the total number of outcomes in the sample space. Where most go wrong is to miscalculate the sample space. Their reasoning runs something like this: “We know that each die has six sides. We have rolled two dice, and so the total number of possible outcomes must be 6 + 6 = 12.”

Although this explanation was straightforward, it is unfortunately incorrect. It’s plausible that going from one die to two should cause us to add six to itself and get 12, but this comes from not thinking carefully about the problem.

### A Second Attempt

Rolling two fair dice more than doubles the difficulty of calculating probabilities. This is because rolling one die is independent of rolling a second one. One roll has no effect on the other one. When dealing with independent events we use the multiplication rule. The use of a tree diagram demonstrates that there are really 6 x 6 =36 outcomes from rolling two dice.

To think about this, suppose that the first die we roll comes up as a 1. The other die could be either a 1, 2, 3, 4, 5 or 6. Now suppose that the first die is a 2. The other die again could be either a 1, 2, 3, 4, 5 or 6. We have already found 12 potential outcomes, and have yet to exhaust all of the possibilities of the first die. A table of all 36 of the outcomes are in the table below.

### Sample Problems

With this knowledge we can calculate all sorts of two dice probability problems. A few follow:

- Two fair six-sided dice are rolled. What is the probability that the sum of the two dice is seven?
- Two fair six-sided dice are rolled. What is the probability that the sum of the two dice is three?
- Two fair six-sided dice are rolled. What is the probability that the numbers on the dice are different?

**SOLUTION:** We can use the table below to see that there are six ways to get a sum of seven with two dice: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1). There are a total of 36 outcomes, so the probability that we’re looking for is 6/36 = 1/6.

**SOLUTION:** We can use the table below to see that there are two ways to get a sum of three with two dice: (1, 2) and (2, 1). There are a total of 36 outcomes, and the probability of rolling a sum of three is 2/36 = 1/18.

**SOLUTION:** The probability that the numbers are the same is 1/6. So the probability that they are *not* the same is 1 - 1/6 = 5/6.

### Three (Or More) Dice

The same principle applies if we are working on problems involving three dice. We multiply and see that there are 6 x 6 x 6 = 216 outcomes. As it gets cumbersome to write the repeated multiplication, we can use exponents to simplify our work. For two dice there are 6^{2} outcomes. For three dice there are 6^{3} outcomes. In general, if we roll *n* dice, then there are a total of 6^{n} outcomes.

## Outcomes for Two Dice

1 | 2 | 3 | 4 | 5 | 6 | |

1 | (1, 1) | (1, 2) | (1, 3) | (1, 4) | (1, 5) | (1, 6) |

2 | (2, 1) | (2, 2) | (2, 3) | (2, 4) | (2, 5) | (2, 6) |

3 | (3, 1) | (3, 2) | (3, 3) | (3, 4) | (3, 5) | (3, 6) |

4 | (4, 1) | (4, 2) | (4, 3) | (4, 4) | (4, 5) | (4, 6) |

5 | (5, 1) | (5, 2) | (5, 3) | (5, 4) | (5, 5) | (5, 6) |

6 | (6, 1) | (6, 2) | (6, 3) | (6, 4) | (6, 5) | (6, 6) |