When I play Settlers of Catan, I notice my opponents often place their initial settlements next to sixes and eights. Why? Because, as everyone knows, when rolling two six-sided dice, sums of six or eight come up more frequently than any other sum (except seven).

However…

### It is a little-known fact that, on any given die roll, the roll is more likely to NOT be a six *or* eight. It is more likely to be anything BUT.

*Here’s the proof:*

0. Make a spreadsheet (or just get a lined piece of paper).

1. In the first column, enter 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 and so on to six 6s. (You should have 36 rows populated.)

2. In the second column, type six sets of 1 2 3 4 5 6.

3. In the third column, sum each row. (1+1=2, 1+2=3, and so on.)

4. Sort the sheet by that third column, so that all the sums are in ascending order. (This just makes the next step easier.)

5. Now, off to the side somewhere, count up the frequency of each sum (how often each sum appears). You should end up with this:

sum frequency

2: 1

3: 2

4: 3

5: 4

6: 5

7: 6

8: 5

9: 4

10: 3

11: 2

12: 1

6. Look again at the first three columns. Notice there are 36 possible outcomes:

36 possible unique rolls

7. Look at the frequency of 6 and 8:

5 possible sixes

5 possible eights

8. Which means there are 26 *non*-six, *non*-eight possibilities.

26 possible everything elses

9. Do the final math that answers this question: how likely is it that a six or eight will be rolled, instead of anything else?

36 possible unique rolls

5 possible sixes

5 possible eights

26 possible everything elses

=

28% probability of 6 or 8

72% probability of NOT 6 or 8

And that, my friends, is why you should stop being surprised that sixes and eights come up so little. On any given roll, you pretty much have only a 1 in 4 chance of it being a six or an eight.

I hope this blows your mind. Does it? Leave a comment and let me know!