Thirty-two players ranked 1 to 32 are playing in 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 p, then the value of `[2//p]` is, where [.] represents the greatest integer function,_____.

Explanation

To solve the problem, we need to find the probability \( p \) that the player ranked 1 wins the tournament and the player ranked 2 is the runner-up. We will do this by ensuring that player ranked 2 never plays against player ranked 1 until the final match.

### Step-by-Step Solution:

1. **Understanding the Tournament Structure**:
In a knockout tournament with 32 players, the players are eliminated in each round until one winner remains. The matches are structured such that the better-ranked player always wins.

2. **Ensuring Player 2 Does Not Face Player 1**:
For player 2 to reach the finals without facing player 1, they must avoid playing against player 1 in all previous rounds.

3. **Calculating the Probability for Each Round**:
- **First Round**: Player 2 can face any of the other 31 players. The probability that player 2 does not face player 1 is \( \frac{30}{31} \) (since there are 30 other players).

- **Second Round**: If player 2 wins their first match, there will be 16 players left. The probability that player 2 does not face player 1 in this round is \( \frac{14}{15} \).

- **Third Round**: If player 2 wins again, there will be 8 players left. The probability that player 2 does not face player 1 is \( \frac{6}{7} \).

- **Fourth Round**: If player 2 wins this round, there will be 4 players left. The probability that player 2 does not face player 1 is \( \frac{2}{3} \).

4. **Calculating the Total Probability**:
The total probability \( P \) that player 2 does not face player 1 until the finals is the product of the probabilities from each round:
\[ P = \left(\frac{30}{31}\right) \times \left(\frac{14}{15}\right) \times \left(\frac{6}{7}\right) \times \left(\frac{2}{3}\right) \]

5. **Simplifying the Expression**:
\[ P = \frac{30 \times 14 \times 6 \times 2}{31 \times 15 \times 7 \times 3} \]
- Calculate the numerator: \( 30 \times 14 = 420 \), \( 420 \times 6 = 2520 \), \( 2520 \times 2 = 5040 \).
- Calculate the denominator: \( 31 \times 15 = 465 \), \( 465 \times 7 = 3255 \), \( 3255 \times 3 = 9765 \).
\[ P = \frac{5040}{9765} \]

6. **Finding \( p \)**:
Since \( p = P \), we have:
\[ p = \frac{5040}{9765} \]

7. **Calculating \( \frac{2}{p} \)**:
\[ \frac{2}{p} = \frac{2 \times 9765}{5040} = \frac{19530}{5040} \]
- Simplifying \( \frac{19530}{5040} \):
\[ \frac{19530 \div 630}{5040 \div 630} = \frac{31}{8} \]

8. **Finding the Greatest Integer Function**:
\[ \left\lfloor \frac{31}{8} \right\rfloor = \left\lfloor 3.875 \right\rfloor = 3 \]

### Final Answer:
The value of \( \left\lfloor \frac{2}{p} \right\rfloor \) is \( 3 \).