Question

Randomly select 24 component and accept the whole batch if there are fewer then 3 defective. If a particular shipment of thousands of components actually has 4% rate of defects. What is the probability that this whole shipment will be accepted?

Answer 1

Let $X=$ the number of defective components: $X\sim Bin(n,p)$

$$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$$

Given that $n=24,p=0.04$

The probability that this whole shipment will be accepted is

$$P(X<3)=P(X=0)+P(X=1)+P(X=2)=$$

$$=\binom{24}{0}0.04^0(1-0.04)^{24-0}+\binom{24}{1}0.04^1(1-0.04)^{24-1}+$$

$$+\binom{24}{2}0.04^2(1-0.04)^{24-2}\approx$$

$$\approx0.375413+0.375413+0.179886\approx0.9307$$