Quality Control on a Production Line
A factory produces circuit boards in batches. Historical records show that 4% of boards are defective, independently of one another. A batch of 20 boards is drawn for inspection. The quality manager must decide whether to accept the batch. Two acceptance rules are on the table: Rule X accepts the batch if zero defectives are found in the sample of 20; Rule Y accepts if at most one defective is found. The manager also has a cheaper option: inspect only 5 boards and accept if zero defectives appear (Rule Z). She wants to compare the probability of acceptance under each rule, and she is uneasy that Rule Z, though cheapest, may accept too many bad batches. She asks for exact probabilities and a clear recommendation.
1. Modelling the number of defectives in a sample of 20 as binomial, compute the acceptance probabilities under Rule X and Rule Y.
Read the worked answer
Let X ~ Binomial(n = 20, p = 0.04).
Rule X: P(accept) = P(X = 0) = 0.96²⁰.
ln(0.96²⁰) = 20 × ln 0.96 = 20 × (−0.040822) ≈ −0.81644, so 0.96²⁰ ≈ 0.4418.
Rule Y: P(accept) = P(X = 0) + P(X = 1).
P(X = 1) = C(20,1)(0.04)(0.96¹⁹) = 20 × 0.04 × 0.96¹⁹.
0.96¹⁹ = 0.96²⁰ / 0.96 ≈ 0.4418 / 0.96 ≈ 0.4602.
P(X = 1) ≈ 0.8 × 0.4602 ≈ 0.3682.
So P(accept under Y) ≈ 0.4418 + 0.3682 = 0.8100.
Interpretation: Rule X rejects about 56% of batches even though the process is in control; Rule Y rejects about 19%.
2. Compute the acceptance probability under Rule Z (5 boards, accept if zero defectives) and compare it with Rule X. What does the comparison reveal about the trade-off between sample size and protection?
Read the worked answer
Under Rule Z, Y ~ Binomial(n = 5, p = 0.04):
P(accept) = 0.96⁵.
0.96² = 0.9216; 0.96⁴ = 0.9216² ≈ 0.84935; 0.96⁵ ≈ 0.84935 × 0.96 ≈ 0.8154.
Comparison:
- Rule X (n = 20, accept on 0): ≈ 0.442
- Rule Z (n = 5, accept on 0): ≈ 0.815
- Rule Y (n = 20, accept on ≤1): ≈ 0.810
Rule Z accepts a good batch about 82% of the time, similar to Rule Y, but it inspects only a quarter as many boards. The catch is protection: with only 5 boards, a batch with, say, a 20% defect rate still passes with probability 0.8⁵ ≈ 0.328, whereas Rule X passes it with probability 0.8²⁰ ≈ 0.012. So Rule Z is cheap but weak at catching bad batches; Rule X is strict but rejects many acceptable batches.
Recommendation: if the priority is screening out deteriorated batches, use Rule X (or Rule Y as a compromise); Rule Z should only be used where inspection cost dominates and defective rates are known to be stable near 4%.