Binomial Probability in Quality Control

00:00 8 marks

In a factory, the quality control department inspects products and classifies them as defective or non-defective. From past data, it is known that 5% of all products are defective. A random sample of 20 products is selected for inspection.

graph TB
    A["Binomial Distribution Parameters"] --> B["n = 20 trials (sample size)"];
    A --> C["p = 0.05 (probability of defect)"]
    A --> D["q = 0.95 (probability of non-defect)"]
    
    E["Calculating Binomial Probability"] --> F["P(X = k) = nCk * p^k * q^(n-k)"]
    E --> G["P(X ≥ k) = 1 - P(X < k)"]
    E --> H["E(X) = n * p"]
    
    style A fill:#d6d7f9,stroke:#030f40
    style B fill:#d6d7f9,stroke:#030f40
    style C fill:#d6d7f9,stroke:#030f40
    style D fill:#d6d7f9,stroke:#030f40
    style E fill:#d6d7f9,stroke:#030f40
    style F fill:#d6d7f9,stroke:#030f40
    style G fill:#d6d7f9,stroke:#030f40
    style H fill:#d6d7f9,stroke:#030f40

Part a

2 minutes 2 marks

Question

Find the probability that exactly 2 products in the sample are defective.

Hint Hint

Hint

Use the binomial probability formula with k = 2.

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Part b

3 minutes 3 marks

Question

Find the probability that at least 3 products in the sample are defective.

Hint Hint

Hint

Consider using the complement method: P(X ≥ 3) = 1 – P(X < 3).

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Part c

3 minutes 3 marks

Question

Calculate the expected number of defective products in the sample of 20.

Hint Hint

Hint

Use the formula for expected value of a binomial distribution.

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