Dice #
The standard six-sided die is used in games of chance like yatzee and farkle.
How to find probability of any dice roll #
Since all possible rolls are equally likely the Probability of Specific Outcome is equal to the Number of Ways to Get Specific Outcome divided by the Number of Ways to Get All Possible Outcomes.
Examples to illustrate dice probabilities: #
Probability of rolling a 6 with one die. #
When you roll a standard six sided die with faces numbered 1 through 6, there are six possible outcomes: (1, 2, 3, 4, 5, 6). There is 1 way to roll a six. Therefore the probability of rolling a six with one die is 1/6 = .1666 = 16.7% about.
\[\text{Probability of a 6} = \frac{\text{1 way to get a six}}{6 \text{ possible outcomes}} = \frac{1}{6}\]Probabiilties of rolling two dice that sum to 6. #
To find the probability of rolling two dice that sum to 6 we need to find two numbers: first, how many ways can two dice be rolled that sum to 6; second, how many rolls are there with two dice.
The table below lists the number of ways to roll two dice that sum to six:
| value of die 1 | value of 2 die |
|---|---|
| 1 | 5 |
| 2 | 4 |
| 3 | 3 |
| 4 | 2 |
| 5 | 1 |
So there are 5 ways to roll two dice that sum to 6. There are a total of 36 possible rolls with two dice. The table below shows all 36 possible rolls with two dice and the 5 rolls that lead to a sum of 6 are bolded.
| die 1 \ die 2 | 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 |
Therefore the probability of rolling two dice that sum to six is 5 / 36 = .138888 = 13.9% about. There is a 5 in 36 chance of rolling two dice that sum to 6.
\[\text{Probability of rolling two dice that sum to six} = \frac{5}{36} \]Probability of rolling two pair with 4 dice. #
To find the probability of rolling two pairs when rolling four dice, we need to divide the number of ways to roll two pair by the total number of rolls possible with four dice. Extending the logic for the number of rolls with two dice (6*6 = 6^2 = 36), the total number of rolls when rolling 4 dice is 6 * 6 * 6 * 6 = 6^4 = 1296. It is more challenging to calculate the number of ways to roll two pairs.
Whenever there are two pair rolled, there are two of the 6 numbers that are paired, for example the two numbers that are paired might be 2 and 4 or 6 and 1. There are 15 possible ways to pick two of the six numbers on the die faces to be pairs:
\[ {6 \choose 2} = \frac{6!}{4!2!} = \frac{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 2 \cdot 1} = \frac{6 \cdot 5}{2} = 15 \]The table below lists all 15 to verify the above math:
| number of ways | ||
|---|---|---|
| 1 | 1 | 2 |
| 2 | 1 | 3 |
| 3 | 1 | 4 |
| 4 | 1 | 5 |
| 5 | 1 | 6 |
| 6 | 2 | 3 |
| 7 | 2 | 4 |
| 8 | 2 | 5 |
| 9 | 2 | 6 |
| 10 | 3 | 4 |
| 11 | 3 | 5 |
| 12 | 3 | 6 |
| 13 | 4 | 5 |
| 14 | 4 | 6 |
| 15 | 5 | 6 |
Next we need to figure out how to arrange the possible pairs. Let’s say we have a pair of 2s and a pair of 4s. There are six possible ways that this pair could arise with the 4 dice rolls.
| 2244 | 2424 | 2442 | 4224 | 4242 | 4422 |
This means that for any two numbers which are paired, there are 6 ways that the two pairs could arise. If the first number paired is “X”, and the second number paired is “Y”, then the ways we can arrange X and Y are below (the same pattern we had above for 2 and 4.)
| XXYY | XYXY | XYYX | YXXY | YXYX | YYXX |
Note that the number of ways to arrange the 4 dice is:
\[ {4 \choose 2} = \frac{4!}{2!2!} = \frac{4 \cdot 3 \cdot 2 \cdot 1}{2 \cdot 1 \cdot 2 \cdot 1} = \frac{4 \cdot 3}{2} = 6 \]To understand why it is 4 choose 2, you can think of one of the letters, say X, as a sort of background and you need to count how many ways you can place the other letter on top of it. There are four possible locations, call them 1, 2, 3, and 4, which the two Y’s can occupy. So the problem is to figure out how many ways you can assign the four locations to the two Y’s hence, 4 choose 2. The same logic would hold if you instead considered Y the background and tried to figure out how many ways you could assign the 4 locations to the 2 X’s.
Putting it all together, the number of ways to get two pair is the number of ways to select two numbers to be paired multiplied by the number of ways to arrange the 4 dice containing two pairs. Thus there are 15 * 6 = 90 ways for two pair to arise when 4 dice are rolled. So the probability of rolling two pair with four dice is 90 / 1296 = 0.069444 = 6.94% about.
\[\text{P(two pair with four dice)} = \frac{90}{1296} = 0.06944 = 6.944\%\]conclusion to examples
This last example was more complex than the first two. Read on to see more examples and gain greater familiarity with the techniques for calculating the probability of dice rolls.