To solve the problem step by step, we will analyze the given information and calculate the required probability.
### Step 1: Understand the problem
We have a positive integer \( n \) that can take values from 1 to 100. The probability of choosing \( n \) is defined differently based on whether \( n \) is less than or equal to 50 or greater than 50.
- If \( n \leq 50 \), the probability of choosing \( n \) is \( p \).
- If \( n > 50 \), the probability of choosing \( n \) is \( 3p \).
### Step 2: Calculate total probabilities
The total number of integers from 1 to 100 is 100. We can split this into two parts:
- For \( n \leq 50 \): There are 50 integers (1 to 50).
- For \( n > 50 \): There are 50 integers (51 to 100).
The total probability must sum to 1:
\[
50p + 50(3p) = 1
\]
\[
50p + 150p = 1
\]
\[
200p = 1 \implies p = \frac{1}{200}
\]
### Step 3: Identify perfect squares
Next, we need to find how many perfect squares are there in the range from 1 to 100:
- The perfect squares less than or equal to 50 are: 1, 4, 9, 16, 25, 36, 49 (which are \( 1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2 \)). There are 7 perfect squares.
- The perfect squares greater than 50 are: 64, 81, 100 (which are \( 8^2, 9^2, 10^2 \)). There are 3 perfect squares.
### Step 4: Calculate probabilities for perfect squares
Now, we calculate the probabilities for choosing a perfect square:
1. **For \( n \leq 50 \)**:
- The probability of choosing a perfect square \( n \) (which is one of the 7 perfect squares) from this range is:
\[
P(\text{perfect square} | n \leq 50) = 7 \cdot p = 7 \cdot \frac{1}{200} = \frac{7}{200}
\]
2. **For \( n > 50 \)**:
- The probability of choosing a perfect square \( n \) (which is one of the 3 perfect squares) from this range is:
\[
P(\text{perfect square} | n > 50) = 3 \cdot (3p) = 3 \cdot 3 \cdot \frac{1}{200} = \frac{9}{200}
\]
### Step 5: Total probability of choosing a perfect square
Now we can find the total probability of choosing a perfect square:
\[
P(\text{perfect square}) = P(\text{perfect square} | n \leq 50) + P(\text{perfect square} | n > 50)
\]
\[
P(\text{perfect square}) = \frac{7}{200} + \frac{9}{200} = \frac{16}{200} = \frac{4}{50} = \frac{2}{25}
\]
### Step 6: Final answer
Thus, the probability that a perfect square is chosen is:
\[
\frac{2}{25} \text{ or } 0.08
\]
