Question

In a factory which manufactures bolts, machines A, B and C manufacture $30\%,50\%$ and $20\%$ of the bolts respectively. Of their output $3\%,4\%$ and $1\%$ respectively are defective bolts. A bolt is drawn at random from the product and is found to be defective. Find the probability that this is not manufactured by machine B.

Answer 1

x $=$ bolt manufactured by company A.

y $=$ bolt manufactured by company B.

z $=$ bolt manufactured by company C.

w $=$ defective bolt drawn.

$P\left(x\right)=\frac{3}{10},P\left(y\right)=\frac{1}{2},P\left(z\right)=\frac{1}{5}$

$P\left(w|x\right)=\frac{3}{100},P\left(w|y\right)=\frac{4}{100},P\left(w|z\right)=\frac{1}{100}$

Probability that defective bolt is drawn from machine B.

$P\left(y|w\right)=\frac{P\left(y\right).P\left(w|y\right)}{P\left(x\right).P\left(w|x\right)+P\left(y\right).P\left(w|y\right)+P\left(z\right).P\left(w|z\right)}$

$P\left(y|w\right)=\frac{20}{31}$

Probability that it is not manufactured by B $=1-\frac{20}{31}=\frac{11}{31}$.