Question

8. The proportion of defective items in a certain manufacturing process is 0.20. If 10
items are selected from the process, Find
a. The expected number of good items
b. Probability that none is defective
c. Probability that at least 3 are defective

Answer 1

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

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

Givven that $p=0.2, n=10$

a. The expected number of good items 

$$P(\bar{X})=n(1-p)=10(1-0.2)=8$$

b. Probability that none is defective 

$$P(X=0)=\binom{10}{0}(0.2)^0(1-0.2)^{10-0}=$$ $$(0.8)^{10}=0.1073741824\approx0.1074$$

c. Probability that at least 3 are defective

$$P(X\geq3)=1-P(X<3)=$$ $$=1-P(X=0)-P(X=1)-P(X=2)=$$ $$=1-\binom{10}{0}(0.2)^0(1-0.2)^{10-0}-$$ $$-\binom{10}{1}(0.2)^1(1-0.2)^{10-1}-\binom{10}{2}(0.2)^2(1-0.2)^{10-2}=$$ $$1-(0.8)^{10}-10(0.2)(0.8)^9-45(0.2)^2(0.8)^8=$$ $$=1-0.1073741824-0.268435456-$$ $$-0.301989888=0.3222004736\approx$$ $$\approx0.3222$$