Science, Tech, Math › Math Probabilities for Rolling Two Dice Share Flipboard Email Print Tetra Images/Getty Images Math Statistics Probability & Games Statistics Tutorials Formulas Descriptive Statistics Inferential Statistics Applications Of Statistics Math Tutorials Geometry Arithmetic Pre Algebra & Algebra Exponential Decay Functions Worksheets By Grade Resources View More by Courtney Taylor Courtney K. Taylor, Ph.D., is a professor of mathematics at Anderson University and the author of "An Introduction to Abstract Algebra." Updated November 04, 2019 One popular way to study probability is to roll dice. A standard die has six sides printed with little dots numbering 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 is equally likely. Since there are six possible outcomes, the probability of obtaining any side of the die is 1/6. The probability of rolling a 1 is 1/6, the probability of rolling a 2 is 1/6, and so on. But what happens if we add another die? What are the probabilities for rolling two dice? Dice Roll Probability To correctly determine the probability of a dice roll, we need to know two things: The size of the sample space or the set of total possible outcomesHow often an event occurs In probability, an event is a certain subset of the sample space. For example, when only one die is rolled, as in the example above, the sample space is equal to all of the values on the die, or the set (1, 2, 3, 4, 5, 6). Since the die is fair, each number in the set occurs only once. In other words, the frequency of each number is 1. To determine the probability of rolling any one of the numbers on the die, we divide the event frequency (1) by the size of the sample space (6), resulting in a probability of 1/6. 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. When dealing with independent events we use the multiplication rule. The use of a tree diagram demonstrates that there are 6 x 6 = 36 possible outcomes from rolling two dice. Suppose that the first die we roll comes up as a 1. The other die roll could be a 1, 2, 3, 4, 5, or 6. Now suppose that the first die is a 2. The other die roll again could be 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. Probability Table of Rolling Two Dice The possible outcomes of rolling two dice are represented in the table below. Note that the number of total possible outcomes is equal to the sample space of the first die (6) multiplied by the sample space of the second die (6), which is 36. 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) 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 possible outcomes. As it gets cumbersome to write the repeated multiplication, we can use exponents to simplify work. For two dice, there are 6^2 possible outcomes. For three dice, there are 6^3 possible outcomes. In general, if we roll n dice, then there are a total of 6^n possible outcomes. Sample Problems With this knowledge, we can solve all sorts of probability problems: 1. Two six-sided dice are rolled. What is the probability that the sum of the two dice is seven? The easiest way to solve this problem is to consult the table above. You will notice that in each row there is one dice roll where the sum of the two dice is equal to seven. Since there are six rows, there are six possible outcomes where the sum of the two dice is equal to seven. The number of total possible outcomes remains 36. Again, we find the probability by dividing the event frequency (6) by the size of the sample space (36), resulting in a probability of 1/6. 2. Two six-sided dice are rolled. What is the probability that the sum of the two dice is three? In the previous problem, you may have noticed that the cells where the sum of the two dice is equal to seven form a diagonal. The same is true here, except in this case there are only two cells where the sum of the dice is three. That is because there are only two ways to get this outcome. You must roll a 1 and a 2 or you must roll a 2 and a 1. The combinations for rolling a sum of seven are much greater (1 and 6, 2 and 5, 3 and 4, and so on). To find the probability that the sum of the two dice is three, we can divide the event frequency (2) by the size of the sample space (36), resulting in a probability of 1/18. 3. Two six-sided dice are rolled. What is the probability that the numbers on the dice are different? Again, we can easily solve this problem by consulting the table above. You will notice that the cells where the numbers on the dice are the same form a diagonal. There are only six of them, and once we cross them out we have the remaining cells in which the numbers on the dice are different. We can take the number of combinations (30) and divide it by the size of the sample space (36), resulting in a probability of 5/6. Continue Reading What Are the Probability Outcomes for Rolling Three Dice? The Probability of Rolling a Yahtzee What Is the Definition of a Sample Space in Statistics? What Is the Probability of the Union of 3 or More Sets? How to Calculate the Probability of a Small Straight in Yahtzee What's the Probability of a Large Straight in Yahtzee in One Roll? Probabilities for Liar's Dice What's the Probability of a Full House in Yahtzee in a Single Roll? What's the Probability of Going to Jail in Monopoly? Here's a Comprehensive Cheat Sheet for Fractions How to Find Conditional Probability How Probability Plays Into the Game of Monopoly Look Up Math Definitions With This Handy Glossary What Is a Bell Curve, Anyway? Definition and Explanation Simulate 10 Million Dice Rolls With Computer Programming What Is the Factorial (!) in Mathematics and Statistics?