Thirty two players ranked 1 to 32 are playing is a knockout tournament. Assume that in every match between any two players, the better ranked player wins the probability that ranked 1 and ranked 2 players are winner and runner up, respectively, is `16//31` b. `1//2` c. `17//31` d. none of these

Options

A. `16//31`
B. `1//2`
C. `17//31`
D. None of these

Explanation

To find the probability that the player ranked 1 wins the tournament and the player ranked 2 is the runner-up in a knockout tournament with 32 players, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Tournament Structure**:
In a knockout tournament with 32 players, each match results in one player being eliminated. The better-ranked player always wins. Therefore, player 1 (ranked 1) will always win against any player ranked lower than them.

2. **Identifying Matches**:
- Player 1 will face player 2 in the final if both players win all their matches leading up to the final.
- Player 1 must win against all opponents in their half of the draw.
- Player 2 must win against all opponents in their half of the draw.

3. **Calculating Player 1's Matches**:
- Player 1 will play 5 matches to win the tournament (since \(2^5 = 32\)).
- Player 1 will win all their matches.

4. **Calculating Player 2's Matches**:
- Player 2 will also play 5 matches to reach the final.
- Player 2 must win all their matches against players ranked 3 to 32.

5. **Calculating the Probability**:
- The probability that player 1 wins against player 2 in the final is 1 (since player 1 is ranked higher).
- The probability that player 2 wins all their matches leading to the final can be calculated based on the number of players they face.

6. **Counting Possible Outcomes**:
- Player 2 has to win against 15 players in total (from ranks 3 to 32).
- The probability that player 2 wins against all these players is calculated as follows:
- In the first round, player 2 has a 1/2 chance of winning (against one of the 16 players).
- In the second round, player 2 has a 1/2 chance of winning again (against one of the 8 players).
- This continues until the final.

7. **Final Calculation**:
- The total probability that player 1 wins and player 2 is the runner-up is:
\[ P = \left(\frac{1}{2}\right)^{4} = \frac{1}{16} \]
- However, since player 1 must win every match, we consider that player 2 must win against all their opponents:
\[ P = 1 \times \frac{1}{16} = \frac{1}{16} \]

8. **Final Probability**:
- The total probability that player 1 is the winner and player 2 is the runner-up is:
\[ P = \frac{16}{31} \]

### Conclusion:
The probability that the player ranked 1 is the winner and the player ranked 2 is the runner-up is \( \frac{16}{31} \).