Question

In a toy-making factory, machine A, B and C manufacture respectively 25%, 35% and 40% of the total toys. Of their output 5%, 4% and 2% respectively are defective toys. A toy is drawn at random from the product. What is the probability that the toy-drawn is defective?

• A. 0.225
• B. 0.345
• C. 0.235
• D. none of these

Answer 1

The correct option is B 0.345

Let $E_1,E_2,E_3$ and A be the events defined as follows.
$E_1=$ the toy is manufactured by machine A
$E_2=$ the toy is manufactured by machine B
$E_3=$ the toy is manufactured by machine C
and A = the toy is defective then
$P\left(E_1\right)=\frac{25}{100}=\frac{1}{4},P\left(E_2\right)=\frac{35}{100}=\frac{7}{20}P\left(E_3\right)=\frac{40}{100}=\frac{2}{5}$
and $\left(\frac{A}{E_1}\right)$ = Probability that the toy drawn is defective given the condition that it is a manufactured by machine $A=\frac{5}{100}$
Similarly, $P\left(\frac{A}{E_2}\right)=\frac{4}{100}andP\left(\frac{A}{E_3}\right)=\frac{2}{100}$
Using law of total probability, we have
$P\left(A\right)=P\left(E_1\right)P\left(\frac{A}{E_1}\right)+P\left(E_2\right)P\left(\frac{A}{E_2}\right)+P\left(E_3\right)P\left(\frac{A}{E_3}\right)=\frac{25}{100}\times \frac{5}{100}+\frac{35}{100}\times \frac{4}{100}+\frac{40}{100}\times \frac{2}{100}=0.345$