10% of tools produced by a certain manufacturing process turn out to be defective. Assuming binomial distribution, the probability of 2 defective in sample of 10 tools chosen at random, is

Options

A. `0.368`
B. `0.194`
C. `0.271`
D. Non of these

Answer

To solve the problem of finding the probability of exactly 2 defective tools in a sample of 10 tools, where 10% of the tools produced are defective, we will use the binomial probability formula.

### Step-by-Step Solution

1. **Identify Parameters**:
- Let \( n = 10 \) (the number of trials, or tools in this case).
- Let \( p = 0.1 \) (the probability of success, which is the probability of a tool being defective).
- Therefore, \( q = 1 - p = 0.9 \) (the probability of failure, which is the probability of a tool being non-defective).

**Hint**: Remember that \( p \) is the probability of getting a defective tool, and \( q \) is the probability of getting a non-defective tool.

2. **Use the Binomial Probability Formula**:
The formula for the probability of getting exactly \( r \) successes (defective tools) in \( n \) trials is given by:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]
Here, we want to find \( P(X = 2) \).

**Hint**: The binomial coefficient \( \binom{n}{r} \) represents the number of ways to choose \( r \) successes from \( n \) trials.

3. **Substitute Values**:
Substitute \( n = 10 \), \( r = 2 \), \( p = 0.1 \), and \( q = 0.9 \) into the formula:
\[ P(X = 2) = \binom{10}{2} (0.1)^2 (0.9)^{10-2} \]

**Hint**: Make sure to calculate \( (0.1)^2 \) and \( (0.9)^8 \) correctly.

4. **Calculate the Binomial Coefficient**:
Calculate \( \binom{10}{2} \):
\[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 \]

**Hint**: Factorials can be simplified by canceling out common terms.

5. **Calculate the Probability**:
Now substitute back into the probability formula:
\[ P(X = 2) = 45 \times (0.1)^2 \times (0.9)^8 \]
Calculate \( (0.1)^2 = 0.01 \) and \( (0.9)^8 \):
\[ (0.9)^8 \approx 0.43046721 \]
Now, multiply:
\[ P(X = 2) = 45 \times 0.01 \times 0.43046721 \approx 45 \times 0.0043046721 \approx 0.19371024 \]

**Hint**: Use a calculator for precise multiplication, especially for powers.

6. **Final Result**:
Rounding \( 0.19371024 \) gives approximately \( 0.194 \).

**Hint**: Always check if your final answer matches one of the given options.

### Conclusion
The probability of getting exactly 2 defective tools in a sample of 10 is approximately \( 0.194 \), which corresponds to option B.