A batch of 115 belts is made by machine A. The probability that a belt made by machine A is defective is 0.02

Question

A batch of 115 belts is made by machine A. The probability that a belt made by machine A is defective is 0.02

(b) Show that the probability that this batch contains at least one defective belt is greater than 0.9

Answer 1

+This is a binomial distribution problem

Let p = probability of a success in this case Pr ( belt fails = .02 Then ( 1- p ) = .98 = Pr ( belt does not fail )

Then Pr belt fails x times in n trials is given by: nCrx ( p )^x ( 1-p )^n-x where nCrx is the number of combinations of x items taken from n items

Also note that the probability at least one belt fails equals 1- Pr ( 0 belts fail in 115 trials ) so we need to calculate that probability

Pr ( 0 fails in 115 ties ) = 115Cr0 ( .02 )⁰ (.98 )¹¹⁵ = 1 (1) (.98)¹¹⁵ = .0979 and 1 - .0979 =

.9021 which is at least .9