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 5%. 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? The probability that none of the LED light bulbs are defective is 0.5987 (Type an integer or a decimal. Round to four decimal places as needed.) 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 . (Type an integer or a decimal. Round to four decimal places as needed.) (1,1) More answer box and then click Check Answer.

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### Quality Control and Probability in Manufacturing

#### Scenario Description
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 5%. Suppose a random sample of 10 LED light bulbs is selected for testing. Below are the questions and solutions related to this scenario.

### Part (a)
**Question:** What is the probability that none of the LED light bulbs are defective?

**Solution:**  
The probability that none of the LED light bulbs are defective is calculated as follows:

\[ P(\text{none defective}) = (1 - 0.05)^{10} \]

Substituting the probability of non-defective bulbs (which is 0.95):

\[ P(\text{none defective}) = 0.95^{10} \approx 0.5987 \]

Therefore, the probability that none of the LED light bulbs are defective is **0.5987**.

### Part (b)
**Question:** What is the probability that exactly one of the LED light bulbs is defective?

**Solution:**  
We use the binomial probability formula to determine the probability of exactly one defective LED bulb:

\[ P(X = 1) = \binom{10}{1} (0.05)^1 (0.95)^{10-1} \]

Where:
- \(\binom{10}{1}\) is the number of ways to choose 1 defective bulb from 10,
- \(0.05\) is the probability of a bulb being defective,
- \(0.95\) is the probability of a bulb being non-defective.

Computing the binomial coefficient and the probabilities:

\[ P(X = 1) = 10 \times 0.05 \times 0.95^9 \]

Calculating \(0.95^9\) and plugging in the values, we get:

\[ P(X = 1) \approx 10 \times 0.05 \times 0.6302 \approx 0.3151 \]

Therefore, the probability that exactly one of the LED light bulbs is defective is approximately **0.3151**.

### Explanation of Tools and Diagrams
- **Binomial Probability Formula:** This is used to calculate the probability of a given number of successes in a fixed number of independent experiments with a constant probability of success.
- **Calculator/Software:**
Transcribed Image Text:### Quality Control and Probability in Manufacturing #### Scenario Description 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 5%. Suppose a random sample of 10 LED light bulbs is selected for testing. Below are the questions and solutions related to this scenario. ### Part (a) **Question:** What is the probability that none of the LED light bulbs are defective? **Solution:** The probability that none of the LED light bulbs are defective is calculated as follows: \[ P(\text{none defective}) = (1 - 0.05)^{10} \] Substituting the probability of non-defective bulbs (which is 0.95): \[ P(\text{none defective}) = 0.95^{10} \approx 0.5987 \] Therefore, the probability that none of the LED light bulbs are defective is **0.5987**. ### Part (b) **Question:** What is the probability that exactly one of the LED light bulbs is defective? **Solution:** We use the binomial probability formula to determine the probability of exactly one defective LED bulb: \[ P(X = 1) = \binom{10}{1} (0.05)^1 (0.95)^{10-1} \] Where: - \(\binom{10}{1}\) is the number of ways to choose 1 defective bulb from 10, - \(0.05\) is the probability of a bulb being defective, - \(0.95\) is the probability of a bulb being non-defective. Computing the binomial coefficient and the probabilities: \[ P(X = 1) = 10 \times 0.05 \times 0.95^9 \] Calculating \(0.95^9\) and plugging in the values, we get: \[ P(X = 1) \approx 10 \times 0.05 \times 0.6302 \approx 0.3151 \] Therefore, the probability that exactly one of the LED light bulbs is defective is approximately **0.3151**. ### Explanation of Tools and Diagrams - **Binomial Probability Formula:** This is used to calculate the probability of a given number of successes in a fixed number of independent experiments with a constant probability of success. - **Calculator/Software:**
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