Eight players P1, P2,. ..... P8 play a knock-out tournament. It is known that whenever the players Pi and Pj play; the player Pi will win if i < j. Assuming that the players are paired at random in each round, what is the probability that the player P4 reaches the final ?
#play 3 sets of knockout tournament
To find the probability that player P4 reaches the final in this knock-out tournament, we need to consider the possible pairings and outcomes in each round.
There are a total of three rounds in the tournament: quarter-finals, semi-finals, and the final.
In the quarter-finals, there are 8 players, including P4. P4 needs to win their match to advance to the semi-finals. Since P4 will always win if they are paired with any player ranked lower than them (i.e., i < 4), there are three possible players that P4 can be paired with: P1, P2, or P3.
The probability of P4 being paired with any of these players in the quarter-finals is 3/7, as there are 7 remaining players (excluding P4) and P4 has an equal chance of being paired with any one of them.
Assuming P4 is paired with one of these three players in the quarter-finals, there are two remaining players from the other half of the bracket. P4 will need to win their semi-final match to reach the final. Again, since P4 will always win if paired with any player ranked lower than them (i.e., i < 4), there are two possible players that P4 can be paired with: one from the remaining half of the bracket and one from the already eliminated players.
The probability of P4 being paired with any of these two players in the semi-finals is 2/5, as there are 5 remaining players (excluding P4 and the player from the other half of the bracket) and P4 has an equal chance of being paired with any one of them.
Finally, if P4 wins their semi-final match, they will reach the final. There are no further pairing options in this case.
Therefore, the probability that P4 reaches the final is calculated as follows:
Probability of P4 reaching the final = Probability of P4 winning in the quarter-finals × Probability of P4 winning in the semi-finals
Probability of P4 reaching the final = (3/7) × (2/5) = 6/35.
Therefore, the probability that player P4 reaches the final in this knock-out tournament is 6/35.