A faulty phone, which assembly line built it?

Question

A factory builds phones on two lines. Line A produces $80\%$ of the phones with a $1\%$ fault rate. Line B produces the remaining $20\%$ with a $6\%$ fault rate. A random phone is found to be faulty.

What is the probability Line B built it?

Accepted Answer

Imagine 1{,}000 phones and count the faulty ones from each line. - Line A: 800 phones, 1\% faulty = 8. - Line B: 200 phones, 6\% faulty = 12. Total faulty phones: 8 + 12 = 20. Of these, 12 came from Line B: P(	ext{Line B} \mid 	ext{faulty}) = \frac{12}{20} = \frac{3}{5} = 0.6. Line B makes only a fifth of the phones, but its fault rate is six times higher, so it is responsible for most of the faults. Takeaway. A small source with a much higher defect rate can dominate the defective pool, so a fault points more strongly to the low-volume, low-quality line.

A faulty phone, which assembly line built it?

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