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Worked solution

Carry a rare cap-defect tail into a box-level distribution

In a factory, a machine automatically cuts out metal caps. It is given that 0.5% of the caps are dented. Find the probability that out of 200 caps, at least 4 are dented.

A box containing 200 of such metal caps are analysed. The box is considered ‘substandard’ if it contains at least 4 dented caps. Calculate the probability that at most 6 out of 300 boxes will be ‘substandard’. marks

[2009/SRJC/II/8(modified)]

CJC · Review Problems · Question 7

Answer

One box: 0.01870.0187 (3 s.f.); at most six substandard boxes: 0.6700.670.

Caps in one box

Under the source independent-defect model, let X∼B(200,0.005)X\sim B(200,0.005). Then q=P(X≥4)=1−P(X≤3)≈0.01868133939q=P(X\geq4)=1-P(X\leq3)\approx0.01868133939. The source prints 0.018681 to 5 significant figures and 0.0187 to 3 significant figures.

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Substandard boxes

Let YY count substandard boxes among 300. For independent boxes with the same defect model, Y∼B(300,q)Y\sim B(300,q). At most six means P(Y≤6)≈0.6701592882=0.670P(Y\leq6)\approx0.6701592882=0.670 to 3 significant figures. The printed calculation uses rounded q=0.018681; the explorer and authored calculation retain the unrounded q until final rounding, with the same printed result.

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Common mistakes

The 0.005 probability belongs to an individual cap; it is not the probability of a substandard box.

Solution notes

The single printed [4] covers both requests. The binomial models assume independent, homogeneous defects and box outcomes; the source does not separately ask for these assumptions.

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