# Dicing with Probability

### an application of base Twelve, by Alice

All numbers in this article are in base twelve; T stands for T and E for eleven; the "dozenal point" is represented by an apostrophe.

If we throw a six-sided die the chance of throwing the number six is simply one in six (assuming, of course, that the die is fair).

One Die (6 outcomes)

Scorewayschance
110'2
210'2
310'2
410'2
510'2
610'2
totals61'0

The game of Craps uses two dice; the player rolls the two dice and adds the numbers shown on the dice to make his score. The least he can score is 2 (with both dice showing "1") and the most 10 (both showing "6").

The scores gained by throwing two dice are shown in the table:

 Die 2 Die 1 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 T 5 6 7 8 9 T E 6 7 8 9 T E 10

There are more ways of throwing a "7" than any other number; of the 30 possible results 6 of them are a "7"; 6 out of 30 is one-sixth, or 0'2, or 20 per gross. (Note how base twelve makes the numbers simple; because of the factor 6.)

Two dice (6^2 = *30 outcomes)

scorewayschanceper gross
210'044
320'088
430'1010
540'1414
650'1818
760'2020
850'1818
940'1414
T30'1010
E20'088
1010'044
totals301'00100

With three dice the scores go from 3 to 16 (3 ones, 3 sixes); note that the number of outcomes for T and E are the most frequent with 23 each; the pobability of scoring T or E is 23/160 or 1 in 8 (0'16 or 16 per gross).

Three dice (6^3=160 outcomes)

scorewayschanceper 1000
310'0088
430'02020
560'04040
6T0'06868
7130'0T0T0
8190'120120
9210'148148
T230'160160
E230'160160
10210'148148
11190'120120
12130'0T0T0
13T0'06868
1460'04040
1530'02020
1610'0088
totals1601'0001000

Footnote by Alice: I prefer an upturned "V" (like an "A" without the bar) for ten.