Answer to Question #218243 in Statistics and Probability for muneeb

Question

Large Consignments of computer components are inspected for
defectives by means of a sampling system. Ten components are
examined and the lot is to be rejected if two or more are found
to be defective. If a consignments contains exactly 10%
defectives.
Find the probability of the consignment by using the technique
of Binomial probability distribution that the consignment is:
i) Accepted
ii) Rejected

Answer 1

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

Given $n=10, p=0.1, q=1-p=1-0.1=0.9.$

i)

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

$$=\dbinom{10}{0}(0.1)^0(0.9)^{10-0}+\dbinom{10}{1}(0.1)^1(0.9)^{10-1}$$

$$=0.7360989291$$

The probability that the consignment is accepted is 0.7360989291.

ii)

$$P(X\geq2)=1-P(X<2)$$

$$=1-\dbinom{10}{0}(0.1)^0(0.9)^{10-0}-\dbinom{10}{1}(0.1)^1(0.9)^{10-1}$$

$$=0.2639010709$$

The probability that the consignment is rejected is 0.2639010709.