Not sure how to set up this problem

Question

Wonder Widgets Inc produces Midget Widgets (MW).. Their data shows that 2% of the output is defective. A sample of 25 MWs is taken. (a) What is the probability that exactly two of this sample are defective? (b) What is the probability that at most two of this sample are defective?

Answer 1

Set this up as the binomial expansion of ( x+y)^25 where x is non defective (98%) and y is defective ( 2%)

To find the probability of exactly 2 defective you would need to evaluate the term x^23y^2

The coefficient of this term is 25 choose 2= (25x24)/2 = 300 So

probability is 300(.98)^23(.02)^2= 7.5%

To find at most 2 defective would need the sum of the probabilities of 0 defective which is the x^25 term and the xy^24 term

Probabilty of x^25 = .98^25=60.3%

Probabilty of 1 defective is 25(.98)^24(.02)= 30.8%

Total is 91.6%