The probability that randomly selected calculator from a store is of brand r is proportional to r, r=1,2,..,6. Further, the probability of a calucltor of brand r being defective is `(7-r )/(21), r=1,2,..,6`. Then the probability that a calculator randomly selected from the store being defective is

Options

A. `(8)/(63)`
B. `(13)/(63)`
C. `(55)/(63)`
D. `(50)/(63)`

Explanation

To solve the problem, we need to find the probability that a randomly selected calculator from the store is defective. Let's break down the solution step by step.

### Step 1: Define the Probability of Selecting Each Brand
The probability of selecting a calculator of brand \( r \) (where \( r = 1, 2, \ldots, 6 \)) is proportional to \( r \). This means we can express the probability as:

\[ P(B_r) = \lambda r \]

where \( \lambda \) is a constant that we need to determine.

### Step 2: Find the Value of \( \lambda \)
Since the total probability must sum to 1, we can write:

\[ \sum_{r=1}^{6} P(B_r) = 1 \]

Substituting our expression for \( P(B_r) \):

\[ \sum_{r=1}^{6} \lambda r = 1 \]

Calculating the sum of the first 6 natural numbers:

\[ \sum_{r=1}^{6} r = \frac{6(6 + 1)}{2} = 21 \]

Thus, we have:

\[ \lambda \cdot 21 = 1 \implies \lambda = \frac{1}{21} \]

### Step 3: Define the Probability of a Defective Calculator
The probability that a calculator of brand \( r \) is defective is given by:

\[ P(D | B_r) = \frac{7 - r}{21} \]

### Step 4: Find the Joint Probability of Selecting a Defective Calculator
We need to find the total probability that a calculator is defective, which can be expressed as:

\[ P(D) = \sum_{r=1}^{6} P(D \cap B_r) = \sum_{r=1}^{6} P(D | B_r) P(B_r) \]

Substituting the expressions we have:

\[ P(D) = \sum_{r=1}^{6} \left(\frac{7 - r}{21}\right) \left(\frac{r}{21}\right) \]

### Step 5: Simplify the Expression
This can be simplified as:

\[ P(D) = \frac{1}{21^2} \sum_{r=1}^{6} (7 - r) r \]

### Step 6: Calculate the Sum
Now we need to calculate the sum \( \sum_{r=1}^{6} (7 - r) r \):

\[ \sum_{r=1}^{6} (7 - r) r = \sum_{r=1}^{6} (7r - r^2) = 7\sum_{r=1}^{6} r - \sum_{r=1}^{6} r^2 \]

Calculating \( \sum_{r=1}^{6} r^2 \):

\[ \sum_{r=1}^{6} r^2 = \frac{6(6 + 1)(2 \cdot 6 + 1)}{6} = \frac{6 \cdot 7 \cdot 13}{6} = 91 \]

Now substituting back:

\[ \sum_{r=1}^{6} r = 21 \quad \text{and} \quad \sum_{r=1}^{6} r^2 = 91 \]

So,

\[ \sum_{r=1}^{6} (7 - r) r = 7 \cdot 21 - 91 = 147 - 91 = 56 \]

### Step 7: Final Calculation
Now substituting back into our probability expression:

\[ P(D) = \frac{1}{21^2} \cdot 56 = \frac{56}{441} = \frac{8}{63} \]

### Conclusion
The probability that a randomly selected calculator from the store is defective is:

\[ \boxed{\frac{8}{63}} \]