Dice Question?

I am working on an idea and my brain has froze. If I roll 6 dice (6 sided) and sum the total of the roll I will get an answer between 6 and 36. How do I figure the probability of rolling each total, 6 through 36?

Thanks,
Kepper

Quote: Kepper

...If I roll 6 dice (6 sided) and sum the total of the roll I will get an answer between 6 and 36. How do I figure the probability of rolling each total, 6 through 36?...

Count the number of ways of making a number and divide by the total number of ways that six dice can be thrown. (The total number of ways that six dice can be thrown is 6*6*6*6*6*6 = 46,656.)

For example: Eight can be made with these arrangements (permutations) of dice: 111113, 111131, 111311, 113111, 131111, 311111, 111122, 111212, 111221, 112112, 112121, 112211, 121112, 121121, 121211, 122111, 211112, 211121, 211211, 212111, and 221111. These are the 21 ways to roll an eight, so the probability of rolling an eight with six dice is 21/46656.

However, counting the numbers of ways is a tiresome process. Instead, you can find the answers at this site.

Each die has six sides, there are six of them, so there are 6^6 or 46,656 possible combinations. There are the following possible ways to make each of these totals (divide the possible ways to make a number, by the total possibilities to get the probability of each):

6: 1 (probability = 1/46,656)
7: 6
8: 21
9: 56
10: 126
11: 252
12: 456
13: 756
14: 1161
15: 1666
16: 2247
17: 2856
18: 3431
19: 3906
20: 4221

21: 4332

22: 4221
23: 3906
22 - 36 continue to mirror the above in decending order

If you want to check your math, the Wizard has posted multiple dice probabilities on the Wizard of Odds website here.

Also, please note that I don't know an easy way to figure out the total combinations, and used the brute force method on a spreadsheet.