# Bayes' Theorem - is the probability quantifiable in this case?

Question 4.49 in Newbold (8. ed)

A company receives large shipments of parts from two sources. Seventy percent of the shipments come from a supplier whose shipments typically contain $$10\%$$ defectives, while the remainder are from a supplier whose shipments typically contain $$20\%$$ defectives. A manager receives a shipment but does not know the source. A random sample of $$20$$ items from this shipment is tested, and $$1$$ of the parts is found to be defective. What is the probability that this shipment came from the more reliable supplier? (Hint: Use Bayes' theorem.)

Using $$P(A)=0.7$$, $$P(B)=0.3$$, $$P(D|A)=0.1$$ and $$P(D|B)=0.2$$ I have made a two-way table and came to the conclusion that $$P(A|D)=0.538$$ and $$P(B|D)=0.461$$. So OK, a little over half of the defective parts come from supplier A (which makes sense, given they provide $$70\%$$ of the parts, even if their defect-rate is lower). However, I don't understand how all this relates to the actual question ("What is the probability that this shipment came from the more reliable supplier?"). From what could I calculate that? Could someone help, please?

The more reliable supplier is $$A$$ because its defection rate is smaller. And, we need the following probability: $$P(A|1D, 19D')=\frac{P(1D,19D'|A)P(A)}{P(1D,19D'|A)P(A)+P(1D,19D'|B)P(B)}$$
And, for example, one of the terms can be calculated as $$P(1D,19D'|A)={20 \choose 1}0.1^10.9^{19}$$