A machine that manufactures automobile parts produces defective parts 13% of the time. If 6 parts produced by this machine are randomly selected, what is the probability that at most 1 of the parts are defective?

Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.

To solve this problem, we can use the binomial probability formula because we are dealing with a fixed number of trials (6 parts) and two possible outcomes for each trial (defective or not defective).

### Given:
- Probability of a defective part, \( p = 0.13 \)
- Probability of a non-defective part, \( q = 1 - p = 0.87 \)
- Number of trials (parts), \( n = 6 \)
- We need to find the probability that at most 1 part is defective.

This is the probability that 0 or 1 part is defective, so we need to calculate:
\[ P(X \leq 1) = P(X = 0) + P(X = 1) \]

### Step 1: Calculate \( P(X = 0) \)
The binomial probability formula is:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]

For \( X = 0 \):
\[ P(X = 0) = \binom{6}{0} (0.13)^0 (0.87)^6 \]
\[ P(X = 0) = 1 \times 1 \times (0.87)^6 \]
\[ P(X = 0) \approx 0.4870 \]

### Step 2: Calculate \( P(X = 1) \)
For \( X = 1 \):
\[ P(X = 1) = \binom{6}{1} (0.13)^1 (0.87)^5 \]
\[ P(X = 1) = 6 \times 0.13 \times (0.87)^5 \]
\[ P(X = 1) \approx 6 \times 0.13 \times 0.4984 \]
\[ P(X = 1) \approx 0.3885 \]

### Step 3: Sum the probabilities
Now, add the two probabilities:
\[ P(X \leq 1) = P(X = 0) + P(X = 1) \]
\[ P(X \leq 1) \approx 0.4870 + 0.3885 = 0.8755 \]

### Final Answer:
The probability that at most 1 of the 6 parts are defective is approximately **0.88** (rounded to two decimal places).

Would you like more details or have any further questions?

Here are some related questions:

1. How do you interpret the result of a binomial probability in a real-world context?
2. What is the formula for the binomial coefficient \( \binom{n}{k} \)?
3. How would the probability change if the defective rate increased to 20%?
4. What is the difference between binomial and normal distributions?
5. How can you use a cumulative binomial probability table?
6. What is the expected number of defective parts in a batch of 6?
7. How does the variance of a binomial distribution relate to the probability of success?
8. What are the conditions required for a binomial distribution to be a good model?

**Tip:** When dealing with binomial probabilities, it's useful to remember that the binomial coefficient \( \binom{n}{k} \) represents the number of ways to choose \( k \) successes out of \( n \) trials.

Learn how to calculate the probability that at most 1 out of 6 automobile parts produced by a machine is defective. This problem uses the binomial probability formula and covers key concepts in probability theory, suitable for students in Grades 10-12.

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