Rolling the dice...
What are the most likely outcomes from rolling a pair of dice?
Assuming we have a standard six-sided die, the odds of rolling a particular value are 1/6. There is an equal probability of rolling each of the numbers 1-6. But, when we have two dice, the odds are not as simple. For example, there's only one way to roll a two (snake eyes), but there's a lot of ways to roll a seven (1+6, 2+5, 3+4).
Let's count how many ways there are to get each value, 2 through 12:
| Outcome | List of Combinations | Total |
|---|---|---|
| 2 | 1+1 | 1 |
| 3 | 1+2, 2+1 | 2 |
| 4 | 1+3, 2+2, 3+1 | 3 |
| 5 | 1+4, 2+3, 3+2, 4+1 | 4 |
| 6 | 1+5, 2+4, 3+3, 4+2, 5+1 | 5 |
| 7 | 1+6, 2+5, 3+4, 4+3, 5+2, 6+1 | 6 |
| 8 | 2+6, 3+5, 4+4, 5+3, 6+2 | 5 |
| 9 | 3+6, 4+5, 5+4, 6+3 | 4 |
| 10 | 4+6, 5+5, 6+4 | 3 |
| 11 | 5+6, 6+5 | 2 |
| 12 | 6+6 | 1 |
If we want to calculate the probability of rolling, say, a five, we need to divide the number of ways to get 5 by the total possible combinations of two dice.
How many total combinations are possible from rolling two dice? Since each die has 6 values, there are \(6*6=36\) total combinations we could get. If you add up the numbers in the total column above, you'll get 36.
So, we can calculate the probabilities of each outcome:
| Outcome | Probability |
|---|---|
| 2 | 1/36 = 2.78% |
| 3 | 2/36 = 5.56% |
| 4 | 3/36 = 8.33% |
| 5 | 4/36 = 11.11% |
| 6 | 5/36 = 13.89% |
| 7 | 6/36 = 16.67% |
| 8 | 5/36 = 13.89% |
| 9 | 4/36 = 11.11% |
| 10 | 3/36 = 8.33% |
| 11 | 2/36 = 5.56% |
| 12 | 1/36 = 2.78% |