Skip to main content

Hypergeometric: 4 batteries from 24

Short answer

Use hypergeometric distribution: (i) 0.2880, (ii) 0.2513, (iii) 0.9986.

A quality control engineer inspects a random sample of 4 batteries from each lot of 24 car batteries that is ready to shipment. If such a lot contain six batteries with slight defects. What are the probabilities that the inspector's sample will contain: i) None of the batteries with defect? ii) At least two of the batteries with defects? iii) At most three of the batteries with defect?

N = 24, K = 6 defective, 18 good, sample n = 4 without replacement, so the number of defectives X is hypergeometric:

P(X = x) = C(6, x) · C(18, 4 − x) / C(24, 4), with C(24, 4) = 10626.

x   C(6,x)·C(18,4-x)      P(X = x)
0   1 × 3060  = 3060      0.2880
1   6 × 816   = 4896      0.4608
2   15 × 153  = 2295      0.2160
3   20 × 18   = 360       0.0339
4   15 × 1    = 15        0.0014

i) None defective

P(X = 0) = 3060 / 10626 = 0.2880

ii) At least two defective

P(X ≥ 2) = 1 − P(0) − P(1) = 1 − 0.2880 − 0.4608 = 0.2513 (Direct: (2295 + 360 + 15) / 10626 = 2670 / 10626 = 0.2513.)

iii) At most three defective

P(X ≤ 3) = 1 − P(X = 4) = 1 − 15/10626 = 1 − 0.0014 = 0.9986

Answer: (i) 0.2880, (ii) 0.2513, (iii) 0.9986

✓ Answer checked by Claude Sonnet 5.5

Related questions

All Probability and Statistics chapter 2 questions →