A box contains 10 sample watches, 2 of which are defective. If 2 are selected at random, the probability that both selected are defective is,

Options

A. `(2)/(25)`
B. `(9)/(20)`
C. `(1)/(25)`
D. `(1)/(45)`

Answer

To solve the problem of finding the probability that both selected watches are defective, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Watches**:
We have a total of 10 sample watches in the box.

2. **Identify the Number of Defective Watches**:
Out of these 10 watches, 2 are defective.

3. **Calculate the Total Ways to Select 2 Watches**:
We need to determine how many ways we can select 2 watches from the total of 10. This can be calculated using the combination formula:
\[ \text{Total ways} = \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 \]

4. **Calculate the Favorable Ways to Select 2 Defective Watches**:
Now, we need to find the number of ways to select 2 defective watches from the 2 defective ones available. This is also calculated using the combination formula:
\[ \text{Favorable ways} = \binom{2}{2} = \frac{2!}{2!(2-2)!} = 1 \]

5. **Calculate the Probability**:
The probability of selecting both defective watches can now be calculated using the formula for probability:
\[ P(\text{both defective}) = \frac{\text{Favorable cases}}{\text{Total cases}} = \frac{1}{45} \]

### Final Answer:
The probability that both selected watches are defective is \(\frac{1}{45}\).

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