Probability of rolling a 6 with two dice

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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?

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 outcomes
• How 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.

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.

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 62 possible outcomes. For three dice, there are 63 possible outcomes. In general, if we roll n dice, then there are a total of 6n possible outcomes.

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.

Whether you’re wondering what your chances of success are in a game or are just preparing for an assignment or exam on probabilities, understanding dice probabilities is a good starting point. Not only does it introduce you to the basics of calculating probabilities, it’s also directly relevant to craps and board games. It's easy to figure out the probabilities for dice, and you can build your knowledge from the basics to complex calculations in just a few steps.

Probabilities are calculated using the simple formula:

Probability = Number of desired outcomes ÷ Number of possible outcomes

So to get a 6 when rolling a six-sided die, probability = 1 ÷ 6 = 0.167, or 16.7 percent chance.

Independent probabilities are calculated using:

Probability of both = Probability of outcome one × Probability of outcome two

So to get two 6s when rolling two dice, probability = 1/6 × 1/6 = 1/36 = 1 ÷ 36 = 0.0278, or 2.78 percent.

The simplest case when you're learning to calculate dice probability is the chance of getting a specific number with one die. The basic rule for probability is that you calculate it by looking at the number of possible outcomes in comparison to the outcome you’re interested in. So for a die, there are six faces, and for any roll, there are six possible outcomes. There is only one outcome you’re interested in, no matter which number you choose.

\text{Probability} = \frac{\text{Number of desired outcomes}}{\text{Number of possible outcomes}}

For the odds of rolling a specific number (6, for example) on a die, this gives:

\text{Probability} = 1 ÷ 6 = 0.167

Probabilities are given as numbers between 0 (no chance) and 1 (certainty), but you can multiply this by 100 to get a percentage. So the chance of rolling a 6 on a single die is 16.7 percent.

If you’re interested in rolls of two dice, the probabilities are still simple to work out. If you want to know the likelihood of getting two 6s when you roll two dice, you are calculating “independent probabilities.” This is because the result of one die doesn’t depend on the result of the other die at all. This essentially leaves you with two separate one-in-six chances.

The rule for independent probabilities is that you multiply the individual probabilities together to get your result. As a formula, this is:

\text{Probability of both} = \text{Probability of outcome one} × \text{Probability of outcome two}

This is easiest if you work in fractions. For rolling matching numbers (two 6s, for example) from two dice, you have two 1/6 chances. So the result is:

\text{Probability} = \frac{1}{6} × \frac{1}{6} = \frac{1}{36}

To get a numerical result, you complete the final division:

\frac{1}{36}=1 ÷ 36 = 0.0278

As a percentage, this is 2.78 percent.

This gets a bit more complicated if you’re looking for the probability of getting two specific different numbers on two dice. For example, if you’re looking for a 4 and a 5, it doesn’t matter which die you roll the 4 with or which you roll the 5 with. In this case, it’s best to just think about it as in the previous section. Out of the 36 possible results, you’re interested in two outcomes, so:

\text{Probability} = \frac{\text{Number of desired outcomes}}{\text{Number of possible outcomes}} = \frac{2}{36} = 0.0556

As a percentage, this is 5.56 percent. Note that this is twice as likely as rolling two 6s.

If you want to know how likely it is to get a certain total score from rolling two or more dice, it’s best to fall back on the simple rule: Probability = Number of desired outcomes ÷ Number of possible outcomes. As before, you determine the total outcome possibilities by multiplying the number of sides on one die by the number of sides on the other. Unfortunately, counting the number of outcomes you’re interested in means a little bit more work.

For getting a total score of 4 on two dice, this can be achieved by rolling a 1 and 3, 2 and 2, or a 3 and 1. You have to consider the dice separately, so even though the result is the same, a 1 on the first die and a 3 on the second die is a different outcome from a 3 on the first die and a 1 on the second die.

For rolling a 4, we know there are three ways to get the outcome desired. As before, there are 36 possible outcomes. So we can work this out as follows:

\text{Probability} = \frac{\text{Number of desired outcomes}}{\text{Number of possible outcomes}} = \frac{3}{36}=0.0833

As a percentage, this is 8.33 percent. For two dice, 7 is the most likely result, with six ways to achieve it. In this case, probability = 6 ÷ 36 = 0.167 = 16.7 percent.