A positive integer 'n' not exceeding 100, is chosen in such a way that if ` n le 50`, then the probability of chossing n is 'p' , and if `n gt 50`, then the probability of choising n is '3p'. The probability that a perfect square is chosen is

Options

A. 0.08
B. 0.065
C. 0.05
D. 0.09

Answer

To solve the problem, we will break it down step by step.

### Step 1: Identify the total number of integers
The positive integers \( n \) that can be chosen range from 1 to 100. Thus, there are a total of 100 integers.

### Step 2: Determine the probabilities for different ranges of \( n \)
- For \( n \leq 50 \), the probability of choosing \( n \) is \( p \).
- For \( n > 50 \), the probability of choosing \( n \) is \( 3p \).

### Step 3: Calculate the total probability
The total number of integers from 1 to 100 is 100. The integers from 1 to 50 are 50, and the integers from 51 to 100 are also 50.

The total probability must sum to 1:
\[ 50p + 50(3p) = 1 \]
\[ 50p + 150p = 1 \]
\[ 200p = 1 \]
\[ p = \frac{1}{200} \]

### Step 4: Count the perfect squares
Next, we need to identify the perfect squares within the ranges:
- Perfect squares less than or equal to 50: \( 1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2 \) which are \( 1, 4, 9, 16, 25, 36, 49 \). This gives us 7 perfect squares.
- Perfect squares greater than 50: \( 8^2, 9^2, 10^2 \) which are \( 64, 81, 100 \). This gives us 3 perfect squares.

### Step 5: Calculate the probabilities of choosing a perfect square
- Probability of choosing a perfect square \( n \leq 50 \):
\[ P(\text{perfect square} | n \leq 50) = \frac{7}{50} \cdot p = \frac{7}{50} \cdot \frac{1}{200} = \frac{7}{10000} \]

- Probability of choosing a perfect square \( n > 50 \):
\[ P(\text{perfect square} | n > 50) = \frac{3}{50} \cdot (3p) = \frac{3}{50} \cdot \frac{3}{200} = \frac{9}{10000} \]

### Step 6: Total probability of choosing a perfect square
Now, we can find the total probability of choosing a perfect square:
\[ P(\text{perfect square}) = P(\text{perfect square} | n \leq 50) + P(\text{perfect square} | n > 50) \]
\[ P(\text{perfect square}) = \frac{7}{10000} + \frac{9}{10000} = \frac{16}{10000} = \frac{4}{2500} = \frac{1}{625} \]

### Final Answer
The probability that a perfect square is chosen is \( \frac{1}{625} \).

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