Question

A factory has three machines $X,Y$ and $Z$ producing $1000,2000$ and $3000$ bolts pet day respectively. The machine $X$ produces $1\%$ defective bolts, $Y$ produces $1.5\%$ and $Z$ produces $2\%$ defective bolts. At the end of a day, a bolt is drawn at random and is found to be defective. What is the probability that this defective bolt has been produced by machine $X$?

Answer 1

$A=$ defective

$P\left(X\right)=\frac{1000}{6000},P\left(Y\right)=\frac{2000}{6000},P\left(Z\right)=\frac{3000}{6000}$

P(A\X)$=0.01$, P(A\Y)$=0.015$, P(A\Z)$=0.02$

$P(X\setminus A)=\frac{P\left(X\right).P(A\setminus X)}{P\left(X\right).P(A\setminus X)+P\left(Y\right).P(A\setminus Y)+P\left(Z\right).P(A\setminus Z)}$

$=\frac{1000\times 0.01}{1000\times 0.01+2000\times 0.015+3000\times 0.02}$

$=\frac{10}{10+30+60}=\frac{1}{10}$