A manufacturing company regularly conducts quality control checks at specified periods on the products it manufactures.​ Historically, the failure rate for LED light bulbs that the company manufactures is 13​%. Suppose a random sample of 10 LED light bulbs is selected. Complete parts​ (a) through​ (d) below.
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Part 1
a. What is the probability that none of the LED light bulbs are​ defective?
The probability that none of the LED light bulbs are defective is 
  
0.2484.
​(Type an integer or a decimal. Round to four decimal places as​ needed.)
Part 2
b. What is the probability that exactly one of the LED light bulbs is​ defective?
The probability that exactly one of the LED light bulbs is defective is

To solve this, we’ll use the binomial probability formula, which applies here since each LED light bulb has two outcomes (defective or not defective) and the probability of failure is constant for each bulb.

The binomial probability formula is:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

where:
- \( n = 10 \) (the number of trials or light bulbs in this case),
- \( k \) is the number of defective bulbs,
- \( p = 0.13 \) (the probability of a bulb being defective),
- \( 1 - p = 0.87 \) (the probability of a bulb not being defective).

### Part (a)
We need to find the probability that none of the light bulbs are defective, so \( k = 0 \).

\[
P(X = 0) = \binom{10}{0} (0.13)^0 (0.87)^{10}
\]

Calculating this:

\[
P(X = 0) = 1 \cdot (0.87)^{10} \approx 0.2484
\]

This matches the provided answer.

### Part (b)
Now, let's find the probability that exactly one of the LED light bulbs is defective, so \( k = 1 \).

\[
P(X = 1) = \binom{10}{1} (0.13)^1 (0.87)^9
\]

1. **Calculate \( \binom{10}{1} = 10 \)** (since there are 10 ways to choose one defective bulb out of 10).
2. Then, **calculate \( (0.13)^1 = 0.13 \)**.
3. Next, **calculate \( (0.87)^9 \approx 0.4228 \)**.
4. Finally, multiply these values:

\[
P(X = 1) = 10 \cdot 0.13 \cdot 0.4228 \approx 0.5496
\]

So, the probability that exactly one of the LED light bulbs is defective is approximately **0.5496**.

Would you like further details, or do you have any questions about these steps?

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Here are five related questions to deepen your understanding:

1. How would you calculate the probability that exactly two of the LED light bulbs are defective?
2. What is the probability that at least one of the LED light bulbs is defective?
3. How would the results change if the sample size were increased to 15 light bulbs?
4. What is the expected number of defective light bulbs in a sample of 10?
5. How does changing the failure rate affect the probabilities calculated here?

**Tip:** The binomial probability formula is useful for any scenario with two possible outcomes (e.g., defective vs. not defective) and a constant probability of success or failure.

A manufacturing company regularly conducts quality control checks at specified periods on the products it manufactures. Historically, the failure rate for LED light bulbs that the company manufactures is 13%. Suppose a random sample of 10 LED light bulbs is selected. Complete parts (a) through (d) below. a. What is the probability that none of the LED light bulbs are defective? b. What is the probability that exactly one of the LED light bulbs is defective?

Explore how to calculate the probabilities of defects in a quality control sample of LED light bulbs, with a known defect rate of 13%. This detailed solution applies the binomial probability formula to assess scenarios with zero or one defective bulb out of a sample of 10. Ideal for college-level or advanced high school students.

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