To solve the problem of identifying the sample space for Ashton's results in three games of checkers, we need to consider all possible outcomes for each game. In checkers, Ashton can either win ([tex]\(W\)[/tex]) or lose ([tex]\(L\)[/tex]) each game. Since there are three games, each game will have two possible outcomes, leading to a combination of these outcomes across the three games.
To generate the sample space, we list all possible sequences of wins and losses over the three games:
1. Ashton wins all three games:
- [tex]\(WWW\)[/tex]
2. Ashton wins the first two games and loses the third game:
- [tex]\(WWL\)[/tex]
3. Ashton wins the first game, loses the second game, and wins the third game:
- [tex]\(WLW\)[/tex]
4. Ashton wins the first game but loses the next two games:
- [tex]\(WLL\)[/tex]
5. Ashton loses the first game and wins the next two games:
- [tex]\(LWW\)[/tex]
6. Ashton loses the first game, wins the second game, and loses the third game:
- [tex]\(LWL\)[/tex]
7. Ashton loses the first two games but wins the third game:
- [tex]\(LLW\)[/tex]
8. Ashton loses all three games:
- [tex]\(LLL\)[/tex]
By listing all possible outcomes for each game, we cover every possible combination of wins and losses over the three games. Therefore, the sample space for Ashton's results is:
[tex]\[ \text{Sample Space} = \{WWW, WWL, WLW, WLL, LWW, LWL, LLW, LLL\} \][/tex]
Each element in the set represents a possible sequence of results for the three games of checkers that Ashton plays with his grandmother.
