Probability of getting a second good item

Question

Following a discussion in a programmer's IRC chat, we had this word problem and couldn't exactly determine what the correct answer was - so I was hoping someone could shed some light and explain how to determine the answer.

There are two chip manufacturers. Manufacturer #1 makes 80% good chips and 20% bad chips. Manufacturer #2 makes 30% good chips and 70% bad chips. You buy 2 chips from the same manufacturer, but you don't know from which manufacturer. The first chip is a good chip, so what are the chances that your second chip will also be a good chip?

Bonus points if you can give the odds of a 3rd chip from the same manufacturer being good.

Answer 1

You have 2 chips from the same manufacturer. Let $A$ be the event of the first chip you test being good, $B$ that of the second. Let $M$ be the event that they come from manufacture #1.

Assuming that there was no bias in selecting the manufacturer, ie: $\mathsf P(M)=\mathsf P(\neg M)=\tfrac 12$ , use Bayes' Rule, and the Law of Total Probability to find:

$$\mathsf P(B\mid A)$$

In terms of the four conditional probabilities:

• $\mathsf P(A,B\mid M)$ : the probability of obtaining two good chips given that they come from manufacturer 1.
• $\mathsf P(A,\neg B\mid M)$ : the probability of obtaining a good and bad chip and picking the good one, given that they come from manufacturer 1.
• $\mathsf P(A,B\mid\neg M)$ : the probability of obtaining two good chips given that they come from manufacturer 2.
• $\mathsf P(A,\neg B\mid\neg M)$ : the probability of obtaining a good and bad chip and picking the good one, given that they come from manufacturer 2.

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Note: The probability depends on the interpretation of "first chip". Is it the first chip to arrive, or the first chip tested.