To determine the sample space for an experiment in which an eight-sided number cube (also known as an octahedron) is rolled twice, we can use a tree diagram or create a table to visualize the outcomes.
Since the cube has 8 faces, when rolling it once, the possible outcomes are:
1, 2, 3, 4, 5, 6, 7, 8.
Rolling the Cube Twice:
When we roll the cube a second time, each outcome from the first roll can be paired with each outcome from the second roll.
Sample Space: You can visualize this using a tree diagram:
-
The first level of branches represents the outcomes of the first roll:
- 1
- 2
- 3
- 4
- 5
- 6
- 7
- 8
-
For each outcome of the first roll, there are 8 branches for the second roll:
- For 1: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8)
- For 2: (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (2,7), (2,8)
- For 3: (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (3,7), (3,8)
- For 4: (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (4,7), (4,8)
- For 5: (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (5,7), (5,8)
- For 6: (6,1), (6,2), (6,3), (6,4), (6,5), (6,6), (6,7), (6,8)
- For 7: (7,1), (7,2), (7,3), (7,4), (7,5), (7,6), (7,7), (7,8)
- For 8: (8,1), (8,2), (8,3), (8,4), (8,5), (8,6), (8,7), (8,8)
Counting the Outcomes: Since there are 8 outcomes for the first roll and 8 outcomes for the second roll, you multiply the number of outcomes:
\[ \text{Total Outcomes} = 8 \times 8 = 64 \]
Thus, the total number of different possible outcomes when rolling the eight-sided number cube twice is 64.