While teaching my kids some basics of probability, I noticed something interesting about the expected value of uniform distributions. Since the expected value is a weighted average of outcomes, it hasn't always seemed very useful to me. For example, the expected value of the roll of a six-sided die is (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 7/2 = 3.5, which not only is never the outcome of rolling a six-sided die but doesn't convey anything about how each outcome has the same likelihood. It only gives us limited information about the range of possible outcomes.
But it does tell us something if we roll more than one die and take the sum of the individual outcomes. To illustrate, here are all possible sums when rolling two dice:
| + | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
And a table of each outcome's frequency:
| Outcome | Occurrences |
|---|---|
| 2 | 1 |
| 3 | 2 |
| 4 | 3 |
| 5 | 4 |
| 6 | 5 |
| 7 | 6 |
| 8 | 5 |
| 9 | 4 |
| 10 | 3 |
| 11 | 2 |
| 12 | 1 |
Seven is the most likely outcome. This is more or less common knowledge, especially among players of board games, but I didn't know what sums were most likely when rolling more dice. As an introductory programming exercise for my oldest kids, I had them do the above tabulation for three and four dice. Here are the results.
Using three dice, the most likely sums are 10 and 11:
| Outcome | Occurrences |
|---|---|
| 3 | 1 |
| 4 | 3 |
| 5 | 6 |
| 6 | 10 |
| 7 | 15 |
| 8 | 21 |
| 9 | 25 |
| 10 | 27 |
| 11 | 27 |
| 12 | 25 |
| 13 | 21 |
| 14 | 15 |
| 15 | 10 |
| 16 | 6 |
| 17 | 3 |
| 18 | 1 |
Using four dice, the most likely sum is 14:
| Outcome | Occurrences |
|---|---|
| 4 | 1 |
| 5 | 4 |
| 6 | 10 |
| 7 | 20 |
| 8 | 35 |
| 9 | 56 |
| 10 | 80 |
| 11 | 104 |
| 12 | 125 |
| 13 | 140 |
| 14 | 146 |
| 15 | 140 |
| 16 | 125 |
| 17 | 104 |
| 18 | 80 |
| 19 | 56 |
| 20 | 35 |
| 21 | 20 |
| 22 | 10 |
| 23 | 4 |
| 24 | 1 |
In other words, while the expected value of rolling one die is 3.5 and not very informative, it does tell us that the most common sum of rolling two dice is 7/2 × 2 = 7. When rolling three dice, the most common sum is 7/2 × 3 = 21/2 = 10.5. That's not one of the possible outcomes, but the two outcomes nearest turn out to be the most common. When rolling four dice, the most common sum is 7/2 × 4 = 14.
The pattern holds for rolling five and six dice as well, predicting sums of 17 at 18 for five dice and 21 for six dice. I haven't proved anything, but a little spot-checking shows the relationship holding for die with more sides, which makes sense given we haven't done anything inherently six-sided. I don't expect the relationship to hold for non-uniform distributions.
While this seems like an easy way to mentally calculate what sums are most likely when rolling arbitrary numbers of dice, there's an easier way: simply look at the range of possible sums and find the middle. Rolling three dice, the possible sums range from 3 to 18 and have a middle of 3 + (18 - 3)÷2 = 10.5. With four dice sums range from 4 to 24 with a middle of 4 + (24 - 4)÷2 = 14.