2.4 Backgammon. Backgammon is a board game for two players in which the playing pieces are moved according to the roll of two dice. Players win by removing all of their pieces from the board, so it is usually good to roll high numbers. You are playing backgammon with a friend and you roll two 6s in your first roll and two 6s in your second roll. Your friend rolls two 3s in his first roll and again in his second row. Your friend claims that you are cheating, because rolling double 6s twice in a row is very unlikely. Using probability, show that your rolls were just as likely as his.

The claim is not fair because:

Probability of roliing two 6s = Probability of rolling two 3s = 1/36

Step-by-step explanation:

The probaility of an event is defined as the number of favorable outcomes divided by the number of total possible events.

P (event E) = number of outcomes for event E / number of possible events

1. As first step, you may draw a table to find the sample space (set of all possible outcomes).

Sample space

The results of the rolling two dice are summarized in this table:

Second roll 1 2 3 4 5 6

First roll

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

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

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

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

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

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

2. Now, in that table, you can observe:

The results (3,3) and (6,6) are highlited.

a) Total number of events: 6 * 6 = 36

b) Nnmber of outcomes for the event rolling two 6s (6,6) : 1

c) Number of outcomes for the event rolling two 3s (3,3) : 1

3) Next, you can calculate the probabilities:

a) Probability rolling two 6s = 1 / 36

b) Probability of rolling two 3s: 1 / 36

4. Conclusion: the probabilities prove that rolling two 6s is just as likely as rolling two 3s.