A box contains 15 transistors, 5 of which are defective. An inspector takes out one transistor at random, examines it for defects, and replaces it. After it has been replaced another inspector does the same thing, and then so does a third inspector. The probability that at least one of the inspectors find a defective transistor is equal to :

Options

A. `1//27`
B. `8//27`
C. `19//27`
D. `26//27`

Answer

To solve the problem, we need to find the probability that at least one of the three inspectors finds a defective transistor. We can approach this by first calculating the probability that none of the inspectors find a defective transistor and then subtracting that from 1.

### Step-by-Step Solution:

1. **Identify the Total and Defective Transistors:**
- Total transistors = 15
- Defective transistors = 5
- Non-defective transistors = 15 - 5 = 10

2. **Calculate the Probability of Selecting a Non-Defective Transistor:**
- The probability of selecting a non-defective transistor (P(non-defective)) is given by:
\[ P(\text{non-defective}) = \frac{\text{Number of non-defective transistors}}{\text{Total number of transistors}} = \frac{10}{15} = \frac{2}{3} \]

3. **Calculate the Probability that All Inspectors Select Non-Defective Transistors:**
- Since the inspectors replace the transistors after checking, the selections are independent. Therefore, the probability that all three inspectors select non-defective transistors is:
\[ P(\text{none defective}) = P(\text{non-defective})^3 = \left(\frac{2}{3}\right)^3 = \frac{8}{27} \]

4. **Calculate the Probability that At Least One Inspector Finds a Defective Transistor:**
- The probability that at least one inspector finds a defective transistor is:
\[ P(\text{at least one defective}) = 1 - P(\text{none defective}) = 1 - \frac{8}{27} \]
- To perform the subtraction:
\[ P(\text{at least one defective}) = \frac{27}{27} - \frac{8}{27} = \frac{19}{27} \]

5. **Final Result:**
- The probability that at least one of the inspectors finds a defective transistor is:
\[ \boxed{\frac{19}{27}} \]