discerete mathematics

Question

h) Suppose Marck (M) and Erick (E) are playing a tennis tournament such that the first person to win two games in a row or who wins a total of three games wins the tournament. Find the number of ways the tournament can proceed

Answer 1

Let M represent that Marck has won a game and E represent Erich has won a game.

 

This is a rough tree diagram of the winners.

The winner of the tournament is in bold italic.

 

                                    Game 1               M

                                                            /    \

                                    Game 2           E     M

                                                        /    \

                                    Game 3       E      M

                                                            /    \

                                    Game 4            E     M

                                                         /    \

                                    Game 5         E    M

 

Suppose Marck wins Game 1.

 

Game 2: If Marck wins the second game then he has won the tournament

              since he has won 2 games in a row.

              Otherwise there is a third game.

 

Game 3: If Erich wins the third game then he has won the tournament

              since he has won 2 games in a row.

              Otherwise there is a fourth game.

 

Game 4: If Marck wins the fourth game then he has won the tournament

              because he has won 2 games in a row and also 3 games total.

              Otherwise there is a fifth game.

 

Game 5: If Erich wins the fifth game then he has won the tournament

              because he has won 2 games in a row and also 3 games total.

              If March wins the fifth game then he has won the tournament

              because he has won 3 games total.

 

If Erich won the first game, in the tree replace M with E and E with M.

This is still only 5 games.

 

It takes a total of 5 game to determine a winner and the possible results

are represented in the tree above.