In a factory of fans, machine A, B and C manufacture respectively 20%, 40% and 40% of total fans. Out of their output 2, 3 and 5 percent are defective fans respectively. A fan is drawn at random from the products then the probability that the drawn fan is defective is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 0.036
Let D be the event that the fan is defective. By the theorem of total probability,
\( \displaystyle P(D)=P(A)P(D|A)+P(B)P(D|B)+P(C)P(D|C) \)
\( \displaystyle =0.2\times 0.02+0.4\times 0.03+0.4\times 0.05 \)
\( \displaystyle =0.004+0.012+0.020 \)
\( \displaystyle =0.036 \)
Shortcut: The overall defective rate is a weighted average of 2%, 3% and 5%, so it must lie between 0.02 and 0.05; only 0.036 does.