A tournament is being run between two teams, A and B. This is a 2 - tournament meaning that the first team to win 2 games is the tournament winner (sometimes called a best-two-out-of-three tournament). For the first game, the probability of A winning is PA wins]-1/3. For all ensuing games the probability that team A wins is PA wins - 1/3 unless team A lost on the previous round in which case PA wins-3/5.

a: What is the probability that the tournament requires the full three games to decide a winner?

b: The tournament concludes after two games. What is the probability that A won?

Step-by-step explanation:

For first game PA = 1/3

For second game PA = 1/3 (If A is not lost in first game)

= 2/5 (If A is lost in first game)

For conclusion of game in three matches:

If A wins, the probability is

PA, PB, PA = 1/3 X 2/3 X 3/5 = 6/45

PB, PA, PA = 2/3 X 3/5 X 1/3 = 6/45

If B wins

PA, PB, PB = 1/3 X 2/3 X 2/5 = 4 / 45

PB, PA, PB = 2/3 X 3/5 X 2/3 = 4 / 15

Total probability of conclusion of game in 3 matches

= 6/45 + 6/45 + 4/45 + 4/15 = 28/45

b)

For the game concluding in 2 matches, the probability are as follows

PA, PA = 1/3 X 1/3 = 1/9

PBPB = 2/3 X 2/5 = 4 / 15

Total probability

= 1/9 + 4/15 = 17/45

So PA = 1/9 / 17/45

= 5/17