8n players `P_(1),P_(2),P_(3)……..P_(8n)` play a knock out tournament. It is known that all the players are of equal strength. The tournament is held in 3 where the players are paired at random in each round. If it is given that `P_(1)` wins in the third round. Find the probability that `P_(2)` looses in the second round.

Answer

To solve the problem, we need to find the probability that player \( P_2 \) loses in the second round given that player \( P_1 \) wins in the third round of a knockout tournament with \( 8n \) players.

### Step-by-Step Solution:

1. **Understanding the Tournament Structure**:
- In a knockout tournament, players are paired randomly in each round.
- Since there are \( 8n \) players, in the first round, all players compete, resulting in \( 4n \) winners advancing to the second round.

2. **Progression of Players**:
- After the first round, \( 4n \) players remain.
- In the second round, these \( 4n \) players will again be paired, resulting in \( 2n \) winners who will advance to the third round.

3. **Condition Given**:
- It is given that \( P_1 \) wins in the third round. This means \( P_1 \) must win in both the second and third rounds.

4. **Finding the Probability**:
- We need to find the probability that \( P_2 \) loses in the second round.
- In the second round, \( 4n \) players compete, and \( 2n \) players will lose.
- Since \( P_1 \) is guaranteed to win in the third round, \( P_2 \) can lose in the second round regardless of \( P_1 \)'s matches.

5. **Favorable Outcomes**:
- The total number of players who can lose in the second round is \( 2n \) (since \( 2n \) players will lose).
- The total number of players who could potentially lose (excluding \( P_1 \), since \( P_1 \) must win) is \( 8n - 1 \).

6. **Calculating the Probability**:
- The probability that \( P_2 \) loses in the second round is given by the ratio of the number of favorable outcomes (i.e., \( P_2 \) losing) to the total number of outcomes (i.e., all players except \( P_1 \)).
- Thus, the probability \( P \) that \( P_2 \) loses in the second round is:
\[ P = \frac{2n}{8n - 1} \]

### Final Answer:
The probability that \( P_2 \) loses in the second round given that \( P_1 \) wins in the third round is:
\[ \frac{2n}{8n - 1} \]