1. The problem asks us to understand the sample space of a tennis match where John and Peter play sets, and the first to win two sets wins the match. 2. The sample space lists all possible sequences of wins (J for John, P for Peter) until one player wins two sets. 3. The tree diagram shows all possible outcomes for each set: - First set: John (J) or Peter (P) wins. - Second set: From each first set outcome, John (J) or Peter (P) can win again. - Third set: If neither player has won two sets after the first two, a third set is played with John (J) or Peter (P) winning. 4. The sample space is written as $S = \{JJ, JPJ, JPP, PJJ, PJP, PP\}$. 5. Explanation of each outcome: - $JJ$: John wins first two sets, match ends. - $JPJ$: John wins first set, Peter wins second, John wins third. - $JPP$: John wins first set, Peter wins next two. - $PJJ$: Peter wins first set, John wins next two. - $PJP$: Peter wins first set, John wins second, Peter wins third. - $PP$: Peter wins first two sets, match ends. 6. The tree diagram visually represents these sequences, branching at each set with possible winners, showing all possible paths until a player wins two sets. This helps understand all possible match outcomes and their sequences.