Question

In a bolt factory, machines $A,B$ and $C$ manufacture $25\%,35\%$ and $40\%$ respectively of the total number of bolts. The percentage of defective bolts among the manufactured bolts is $5\%$ for $A$ , $4\%$ for $B$ and $2\%$ for $C$. A bolt is drawn randomly from the manufactured products and is found to be defective then,

• A. the probability that the selected bolt is defective $=0.0345$
• B. the probability that the defective bolt was manufactured by machine $A=\frac{25}{69}$
• C. the probability that the defective bolt was manufactured by machine $B=\frac{28}{69}$
• D. the probability that the defective bolt was manufactured by machine $C=\frac{16}{69}$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A the probability that the selected bolt is defective $=0.0345$
B the probability that the defective bolt was manufactured by machine $C=\frac{16}{69}$
C the probability that the defective bolt was manufactured by machine $A=\frac{25}{69}$
D the probability that the defective bolt was manufactured by machine $B=\frac{28}{69}$

Let $X$ be the event that a defective bolt is manufactured.
$P\left(X\right)=0.25\times 0.05+0.35\times 0.04+0.40\times 0.02=0.0345$
$P\left(A|X\right)=\frac{0.25\times 0.05}{0.0345}=\frac{25}{69}$
$P\left(B|X\right)=\frac{0.35\times 0.04}{0.0345}=\frac{28}{69}$
$P\left(C|X\right)=\frac{0.40\times 0.02}{0.0345}=\frac{16}{69}$