Rolling Two Dice - Maple Help

Rolling Two Dice

Main Concept

Imagine that you are playing some kind of game where you roll two fair dice. What is the chance that you roll two 6's?

To answer this question, first consider the set of all possible outcomes of the experiment of rolling two dice. While the outcome of rolling two dice is not certain, the list of possible outcomes of rolling two dice is known. When we roll two dice, either of the dice can have values of 1 through 6, which results in the following list of outcomes:

{ (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) }

This list contains all of the possible outcomes of the experiment and is referred to as the sample space of the experiment. You can also refer to certain subsets of the sample space, say rolling two 6's, or rolling both even numbers, as events.

From this list, you can observe that for rolling two dice, there are  = ${36}$ possible outcomes and furthermore, you can observe that the outcome (6, 6) only occurs once in this list, meaning that the probability of rolling two dice is $\frac{1}{36}$.

There are many kinds of events that you could be interested in when rolling dice. It can be beneficial to create a table of possible dice-roll outcomes to quickly observe the probabilities of certain events, such as the probability of rolling a specific sum:

 Outcome (Sum) List of Combinations Probability $2$ 1+1 1/36 3 1+2, 2+1 2/36 or 1/18 4 1+3, 2+2, 3+1 3/36 or 1/12 $5$ 1+4, 2+3, 3+2, 4+1 4/36 or 1/9 $6$ 1+5, 2+4, 3+3, 4+2, 5+1 5/36 $7$ 1+6, 2+5, 3+4, 4+3, 5+2, 6+1 6/36 or 1/6 $8$ 2+6, 3+5, 4+4, 5+3, 6+2 5/36 $9$ 3+6, 4+5, 5+4, 6+3 4/36 or 1/9 $10$ 4+6, 5+5, 6+4 3/36 or 1/12 $11$ 5+6, 6+5 2/36 or 1/18 $12$ 6+6 1/36

In the following example, you can select certain outcomes for rolling two dice and view the probabilities of attaining those events.



 Left Die: Choose123456EvenOdd Right Die: Choose123456EvenOdd Both: ChooseSum to 2Sum to 3Sum to 4Sum to 5Sum to 6Sum to 7Sum to 8Sum to 9Sum to 10Sum to 11Sum to 12SameDifferent Either Die is: Choose123456EvenOdd