# probability of exactly one banjo in a case of 6 is defective

#### palmer69

##### New member
If 8% of the banjos that are shipped are defective, and there are 6 banjos per case which is a sampling. What is the probability that exactly one out of a case is defective.

I figured out that 0.48 of the banjos in a case would be defective, and I know that the percentage of exactly one being defective would be less than 8%, but how do I figure out what it would be?

Lori

#### tkhunny

##### Moderator
Staff member
Re: probability of exactly one

You should be able to construct Binomial probabilities.

Pr(Defective) = p = 0.08

Pr(Not Defective) = q = 1-p = 1-0.08 = 0.92

Expand: $$\displaystyle (p+q)^{6}$$

Pr(All Six Defective) = $$\displaystyle p^{6}$$

Pr(Five Defective) = $$\displaystyle 6p^{5}q$$

Pr(Four Defective) = $$\displaystyle 15p^{4}q^{2}$$

Pr(Three Defective) = $$\displaystyle 20p^{3}q^{3}$$

Pr(Two Defective) = $$\displaystyle 15p^{2}q^{4}$$

Pr(One Defective) = $$\displaystyle 6pq^{5}$$

Pr(None Defective) = $$\displaystyle q^{6}$$

Your task is to remember where the coefficients come from. 1 6 15 20 15 6 1 -- This should be very familiar.

Your next task would be to pull one of those terms out of your hat without calculating ALL of them.