To solve the problem, we need to analyze the knockout tournament involving 8n players, where we are given that player P1 wins in the third round. We need to find the probability that player P2 loses in the second round.
### Step-by-Step Solution:
1. **Understanding the Tournament Structure**:
- In a knockout tournament, players are paired randomly in each round, and only the winners advance to the next round.
- With 8n players, in the first round, there will be 4n matches, resulting in 4n winners who advance to the second round.
2. **Second Round Players**:
- In the second round, the 4n players will again be paired randomly into 2n matches, leading to 2n winners who will advance to the third round.
3. **Condition Given**:
- We know that P1 wins in the third round. This means P1 must win in both the first and second rounds to reach the third round.
4. **Analyzing P2's Situation**:
- We need to find the probability that P2 loses in the second round. For P2 to lose in the second round, P2 must win in the first round but lose in the second round.
5. **Calculating the Probability**:
- In the second round, there are 2n players competing, and only 2n/2 = n players will win and advance to the third round.
- Since all players are of equal strength, the probability of any specific player winning or losing is equal.
6. **Favorable Outcomes**:
- The favorable outcome for P2 losing in the second round occurs when P2 is paired against another player and loses. Since there are 2n players in the second round, P2 has a 1/2 chance of losing against any opponent.
7. **Total Outcomes**:
- The total outcomes for P2 in the second round are that P2 can either win or lose, which are both equally likely.
8. **Final Probability Calculation**:
- The probability that P2 loses in the second round, given that P1 wins in the third round, can be expressed as:
\[
P(\text{P2 loses in second round}) = \frac{1}{2}
\]
### Conclusion:
The probability that P2 loses in the second round, given that P1 wins in the third round, is \( \frac{1}{2} \).
