Question

Find the sample space for a roll of a pair of dice (2 six-sided number cubes).

1. A roll of one die has six possible outcomes. Use the product counting principle to determine the total number of outcomes for a toss of two dice. Explain your response.


2. Draw a table to show the sample space of a toss of two dice. Denote the event of a sum of 7 or 11 on the table.



3. What is the probability of a sum of 7 or 11?

Answer 1

1. 36 outcomes

2. outcomes for 7 is (6,1) (5,2) (4,3) (3,4) (2,5) (1,6)

out comes for 11 are (6,5) and (5,6)

3.The probability for 7 is 1/6.

The probability for 11 is 1/18

together they are 2/9

Explanation:

1/6 + 1/18 = 2/9

Answer 2

For one die there are 6 different possible outcomes: 1, 2, 3, 4, 5 or 6.

1. For two dice there are 6·6=36 possible outcomes (see table for details):

`\begin{array}{ccccccc} \text{Sides of dice} & 1 & 2 & 3 & 4 & 5 & 6 \\ 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) \end{array}`.

In this table in pair (a,b) a is number on the first die and b is number on the second die.

2. To roll 7 in total you should consider the largest diagonal, where outcomes are (6,1), (5,2), (4,3), (3,4), (2,5), (1,6).

To roll 11 in total you should consider the smallest diagonal, where outcomes are (6,5), (5,6).

3. The probability to roll 7 in total is 6/36=1/6. The probability to roll 11 in total is 2/36=1/18. The probability of a sum of 7 or 11 is 1/6+1/18=4/18=2/9.