Let $E$ be the event that a smartphone of this model is defective. Let $F_A$ be the event that a smartphone is manufactured by factory A. Similarly for $F_B$ and $F_C$.
Then the overall fraction of defective smartphones of this model can be found as follows.
P(E) &= P(F_A \cap E) + P(F_B \cap E) + P(F_C \cap E)\\
&= P(F_A)\cdot P(E \mid F_A) + P(F_B)\cdot P(E \mid F_B) + P(F_C)\cdot P(E \mid F_C)\\
&= (0.6)(0.05) + (0.25)(0.02) + (0.15)(0.07)\\
Thus, the overall defective rate is $4.55\%$.
In the context of the above problem, if a smartphone of this model is found out to be detective, find the probability that this smartphone was manufactured in factory C.
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