Calculating Probability

Most people would say that the probability of an outcome of heads or tails in a fair coin toss is 50%; some might say it is half, 0.5, or one chance in two. They all would be correct. There is no chance (that is, 0 probability) that neither heads nor tails will come up; there is 100% probability (that is, 1) that either heads or tails will come up.

What if the outcome was the value showing on a die from a pair of dice that was rolled or tossed? Again most people would say that the probability is 1/6 or one chance in six that a single number like a "5" would come up. If two dice were tossed, what is the chance that the sum of the showing faces will equal "7"? This is a little harder problem, but craps players know the answer. We reason this way: There are 36 combinations of faces that could come up with the repetitive roll of the die, like a "1" on both faces (1,1) or a "1" on one face and a "3" on the other (1,3). There are six combinations that total "7" (1,6; 6,1; 2,5; 5,2; 3,4; 4,3) out of a possible 36, so the chances of a "7" are 6/36 or 1/6.

The probability that no combination of any numbers will show up on a roll of the dice is 0; the probability that any combination of numbers will show up is 36/36, or 1. After all, there are 36 ways that any number could show up. Any specific combination like a (1,6) or a (5,3) will show up with probability of 1/36 since a specific combination can only show up one way. Finally, any specific sum of the dice will show up with probability between 1/36 and 6/36. Table 2-1 illustrates this opportunity space.

Table 2-1: Roll of the Dice
Combination Number	Face Value #1	Face Value #2	Sum of Face Values
1	1	1	2
2	2	1	3
3	3	1	4
4	4	1	5
5	5	1	6
6	6	1	7
7	1	2	3
8	2	2	4
9	3	2	5
10	4	2	6
11	5	2	7
12	6	2	8
13	1	3	4
14	2	3	5
15	3	3	6
16	4	3	7
17	5	3	8
18	6	3	9
....	....	....	....
36	6	6	12
Notes: The number "7" is the most frequent of the "Sum of Face Values," repeating once for each pattern of 6 values of the first die, for a total of 6 appearances in 36 combinations.
Although not apparent from the abridged list of values, the number "5" appearing in the fourth-down position in the first pattern moves up one position each pattern repetition and thus has only 4 total appearances in 36 combinations.