The dice game in the casino

When three fair dice are thrown, there are altogether 6 × 6 × 6 = 216 different possible outcomes.

In the casino, if the total sum of three dice is 11 or above, it is called ‘big’. If the total sum is smaller than 11, it is called ‘small’. However, the combinations in which all three numbers are the same will be excluded. There are a total of 105 combinations which are small, listed as follows.

(1, 1, 2), (1, 1, 3), (1, 1, 4), (1, 1, 5), (1, 1, 6)

(1, 2, 1), (1, 2, 2), (1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 2, 6)

(1, 3, 1), (1, 3, 2), (1, 3, 3), (1, 3, 4), (1, 3, 5), (1, 3, 6)

(1, 4, 1), (1, 4, 2), (1, 4, 3), (1, 4, 4), (1, 4, 5)

(1, 5, 1), (1, 5, 2), (1, 5, 3), (1, 5, 4)

(1, 6, 1), (1, 6, 2), (1, 6, 3)

(2, 1, 1), (2, 1, 2), (2, 1, 3), (2, 1, 4), (2, 1, 5), (2, 1, 6)

(2, 2, 1), (2, 2, 3), (2, 2, 4), (2, 2, 5), (2, 2, 6)

(2, 3, 1), (2, 3, 2), (2, 3, 3), (2, 3, 4), (2, 3, 5)

(2, 4, 1), (2, 4, 2), (2, 4, 3), (2, 4, 4)

(2, 5, 1), (2, 5, 2), (2, 5, 3)

(2, 6, 1), (2, 6, 2)

(3, 1, 1), (3, 1, 2), (3, 1, 3), (3, 1, 4), (3, 1, 5), (3, 1, 6)

(3, 2, 1), (3, 2, 2), (3, 2, 3), (3, 2, 4), (3, 2, 5)

(3, 3, 1), (3, 3, 2), (3, 3, 4)

(3, 4, 1), (3, 4, 2), (3, 4, 3)

(3, 5, 1), (3, 5, 2)

(3, 6, 1)

(4, 1, 1), (4, 1, 2), (4, 1, 3), (4, 1, 4), (4, 1, 5)

(4, 2, 1), (4, 2, 2), (4, 2, 3), (4, 2, 4)

(4, 3, 1), (4, 3, 2), (4, 3, 3)

(4, 4, 1), (4, 4, 2)

(4, 5, 1)

(5, 1, 1), (5, 1, 2), (5, 1, 3), (5, 1, 4)

(5, 2, 1), (5, 2, 2), (5, 2, 3)

(5, 3, 1), (5, 3, 2)

(5, 4, 1)

(6, 1, 1), (6, 1, 2), (6, 1, 3)

(6, 2, 1), (6, 2, 2)

(6, 3, 1)

The probability of small is thus . The rate of division is however 1 to 2 only.