In general, the probability of an event occurring is the number
of successes divided by the total number of possible outcomes
(known as the sample space) – assuming that each outcome is
equally likely.
For example, given a six-sided die the probability of rolling a 2
or a 5 is 2/6 because there are six possibilities, of which, two
are “successes.”
However, when rolling two dice, the probability of rolling a
sum of 2 is NOT 1/11. (There are 11 outcomes for the sum, 2
through 12.) This is because each of these possible sums, 2
through 12, are not equally likely.
The real sample space is the 36 ordered pairs (x,y) where 1 ≤
x,y ≤ 6.
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
For example, the probability of rolling a 2 is 1/36, since only
one of these 36 possibilities adds up to 2. The probability of
rolling a 3 is 2/36, a 4 is 3/36, etc. (Seven is the most frequent