Answer to Question #182343 in Statistics and Probability for Sean

Question

new machine is functioning well, 92% of the items produced are non- defective. Assume that a random sample of five parts produced on the machine is selected and we are interested in the number of defective parts found. Required: a) What are the conditions for this experiment to be considered a binomial experiment? Motivate. What is the probability that all parts selected will be defective? (3) c) What is the probability that none of the parts selected will be defective? (3) d) What is the probability that more than five parts will be defective? (2) e) What is the probability that at least three parts will be non-defective?

Answer 1

Let p denote the probability that item is defective $p=1-0.92=0.08$

And q denote the probability that items are non-defective $q=0.92$

n=5

(a) The number of trials should be small, probability of defective items 0.08

Binomal distribution: X~(N,p)~(5,0.08)

(b) Probability all parts defective $P(X=5)=^5C_5P^5q^0$

$=1.(0.08)^5.(0.92)^0=0.0003276$

(c) Probability that none of the parts is defective $P(X=0)=^5C_0P^0q^5$

$=1.(0.08)^0.(0.92)^5=0.65908$

(d) Probability that more than five parts are defective $P(X>5)=0$

(e) Probability at least three parts are defective $P(X\ge 3)=P(X=3)+P(X=4)+P(X=5)\\=^5C_3P^3q^2+^5C_4P^4q^1+^5C_5P^5q^0$ $=10.(0.08)^3.(0.92)^2+5.(0.08)^4.(0.92)^1+1.(0.08)^5.(0.92)^0\\=0.00433+0.000188+0.0003276=0.004846$