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My attempt:

G1: First screw is good. G2: Second screw is good

G1A: First screw comes from A. G1B: First screw comes from B

G2A: Second screw comes from A. G2B: Second screw comes from B

P(G1A)=1/2, P(G2A)=1/2, P(G1B)=1/2, P(G2B)=1/2

P(G1A $\cap$ G2A)=P(G1B $\cap$ G2B)=P(G1A $\cap$ G2B)=P(G1B $\cap$ G2A)=1/2*1/2=1/4

Now,

P(G1)=P(G1|G1A)P(G1A)+P(G1|G1B)P(G1B)

P(G1 $\cap$ G2)=P(G1 $\cap$ G2|G1A $\cap$ G2A)P(G1A $\cap$ G2A)+P(G1 $\cap$ G2|G1B $\cap$ G2B)P(G1B $\cap$ G2B)+P(G1 $\cap$ G2|G1A $\cap$ G2B)P(G1A $\cap$ G2B)+P(G1 $\cap$ G2|G1B $\cap$ G2A)P(G1B $\cap$ G2A)

P(G2|G1)=P(G1 $\cap$ G2)/P(G1).

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Our teacher's answer:

G1: First screw is good. G2: Second screw is good

A: Screw is from factory A B: Screw is from factory B

P(A)=1/2, P(B)=1/2

P(G1)=P(G1|A)P(A)+P(G1|B)P(B)

P(G1 $\cap$ G2)=P(G1 $\cap$ G2|A)P(A)+P(G1 $\cap$ G2|B)P(B)

Now P(G2|G1)=P(G1 $\cap$ G2)/P(G1).

Which answer is correct?