Question

20% of items produced from a factory are defective. What is the probability that in a sample of 5 chosen at random, none is defective, one is defective and p (1<x<4)?

Answer 1

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

Given $n=5, p=0.2$

i)

$$P(X=0)=\dbinom{5}{0}(0.2)^0(1-0.2)^{5-0}=0.32768$$

ii)

$$P(X=1)=\dbinom{5}{1}(0.2)^1(1-0.2)^{5-1}=0.4096$$

iii)

$$P(1<X<4)=P(X=2)+(X=3)$$

$$=\dbinom{5}{2}(0.2)^2(1-0.2)^{5-2}+\dbinom{5}{3}(0.2)^3(1-0.2)^{5-3}$$

$$=0.2048+0.0512=0.256$$