Electric motors coming off two assembly lines are pooled for storage in a common stockroom, and the room contains an equal number of motors from each line. Motors are periodically sampled from that room and tested. It is known that 10% of the motors from line I are defective and 15% of the motors from line II are defective. If a motor is randomly selected from the stock-room and found to be defective, find the probability that it came from line I.
Here is my way to solve it. First it is a conditional probability. The formula is
$P(A \mid B) = \frac{P (A\cap B) }{ P(B) }.$
$P(B)$ = probability that it came from line 1 = $2 P_1$.
Now here is where it gets interesting. What would be $P(A\cap B)$ in that case? Is $P(A \cap B)=P(\text{came from line 1 * defective})$?