Suppose 3 factories, $F_1$, $F_2$, and $F_3$, make bolts, with the probability of a bolt being made in each factory being $P(F_1)=4/7$, $P(F_2)=2/7$ and $P(F_3)=1/7$ respectively.
Bolts produced by Factory 1 are defective independently with probability 0.02, so, $P(D|F_1)=0.02$. Bolts produced by Factory 2 are defective independently with probability 0.02, so, $P(D|F_2)=0.02$. Bolts produced by Factory 3 are defective independently with probability 0.05, so, $P(D|F_3)=0.05$.
Given that two bolts are selected from a batch of bolts from a particular factory and found to be defective, I'm trying to find the probability that the third bolt selected from this batch is also defective. I presume that I need to compute $P(D_3|D_1 \cap D_2)$.
I've attacked the problem as follows, but it doesn't seem to be leading me anywhere helpful:
$$P(D_3|D_1 \cap D_2) = \frac {P(D_3\cap D_1 \cap D_2)}{P(D_1\cap D_2)}$$ $$ = \frac {P(D_1|D_2 \cap D_3)P(D_2|D_3)P(D_3)}{P(D_1 \cap D_2|F_1)P(F_1)+P(D_1 \cap D_2|F_2)P(F_2) + P(D_1 \cap D_2|F_3)P(F_3)} $$