Question

A worn machine is known to produce 10% defective components. If the random

variable X is the number of defective components produced in a run of 4 components, find the probabilities that X takes the values 0 to 4.

Answer 1

$p =0.1 \text{ probability of receiving a defective component}$

$q= 1-p=0.9 \text{ probability of getting the right component}$

$p\text{ independent random variables and } p+q=1$

$\text{hence we have binomial distribution:}$

$p_{x}(k)=\binom{n}{k}p^kq^{n-k}$

$n=4 \text{ by the condition of the problem}$

$p_x(k); k =0,1,2,3,4$

$p_{x}(0)=\binom{4}{0}p^0q^{4}=0.9^4=0.6561$

$p_{x}(1)=\binom{4}{1}p^1q^{3}=4*0.1*0.9^3=0.2916$

$p_{x}(2)=\binom{4}{2}p^2q^{2}=6*0.1^2*0.9^2=0.0486$

$p_{x}(3)=\binom{4}{3}p^3q^{1}=4*0.1^3*0.9=0.0036$

$p_{x}(4)=\binom{4}{4}p^4q^{0}=0.1^4=0.0001$

Answer: 0.6561 probability of getting 0 defective components out of 4

0.2916 probability of getting 1 defective components out of 4

0.0486 probability of getting 2 defective components out of 4

0.0036 probability of getting 3 defective components out of 4

0.0001 probability of getting 4 defective components out of 4