Inspecting incoming car parts from suppliers is a crucial step in automobile industry quality
control procedures. Let's say they received a shipment of 50 car spoiler parts. Each part has
a probability of 0.1 being defective. After thoroughly examining them, they'll return the
whole shipment if 20% or more defectives were found.
b. What is the probability of finding at most 5 defects per shipment?

To find the probability of finding at most 5 defective car spoiler parts in the shipment, we can model this problem using a **binomial distribution** because we have a fixed number of trials (50 parts), each part is independent, and there are two possible outcomes: defective or not defective.

### Binomial Distribution Formula:
The binomial probability formula is given by:

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

Where:
- \( P(X = k) \) is the probability of having exactly \( k \) defects.
- \( n \) is the number of trials (in this case, 50).
- \( k \) is the number of defective parts.
- \( p \) is the probability of a part being defective (0.1).
- \( \binom{n}{k} \) is the binomial coefficient, or "n choose k."

### Our Parameters:
- \( n = 50 \)
- \( p = 0.1 \)
- We are asked to find the probability of having at most 5 defective parts: \( P(X \leq 5) \), meaning we need to sum the probabilities for 0, 1, 2, 3, 4, and 5 defects.

\[
P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
\]

We can use cumulative binomial probabilities to find this sum, either through tables or computational tools. Let me calculate it for you.The probability of finding at most 5 defective car spoiler parts in the shipment is approximately **0.616**, or about **61.6%**.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. What is the probability of finding exactly 5 defective parts?
2. How does the probability change if the shipment size increases to 100 parts?
3. What is the probability of finding more than 10 defective parts in the shipment?
4. How does the threshold of rejecting the shipment (20%) affect the overall probability of rejection?
5. What is the expected number of defective parts in a shipment?

**Tip:** When working with binomial distributions, always check if the conditions for binomial probability are satisfied—fixed number of trials, independent events, and two possible outcomes per trial.

Inspecting incoming car parts from suppliers is a crucial step in automobile industry quality control procedures. Let's say they received a shipment of 50 car spoiler parts. Each part has a probability of 0.1 being defective. After thoroughly examining them, they'll return the whole shipment if 20% or more defectives were found. What is the probability of finding at most 5 defects per shipment?

Discover how to calculate the probability of finding at most 5 defective car spoiler parts in a shipment of 50 parts using binomial distribution. Explore key concepts such as the binomial probability formula, theorems, and a step-by-step calculation. Suitable for high school and undergraduate students.

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