The two-chip module

Question

A factory makes modules that each contain two chips. Of all modules, 40% are type AA (both chips are grade A) and 60% are type AB (one grade A chip and one grade B chip). You grab a random module, pull out one of its two chips at random, and test it. It is grade A.

What is the probability that the module is type AA?

Accepted Answer

Here the two module types are not equally common, so weight by the base rates. A grade-A chip is certain from an AA module and half-likely from an AB module: P(	ext{A chip}) = 0.4\cdot 1 + 0.6\cdot 0.5 = 0.4 + 0.3 = 0.7. By Bayes' rule, P(	ext{AA} \mid 	ext{A chip}) = \frac{0.4\cdot 1}{0.7} = \frac{4}{7} \approx 0.571. A concrete count makes it tangible. Among 100 modules: 40 are AA, contributing 80 grade-A chips, and 60 are AB, contributing 60 grade-A chips (and 60 grade-B). Of the 140 grade-A chips, 80 come from AA modules, so the probability is 	frac{80}{140} = 	frac47. Takeaway. With…

The two-chip module

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