What is the P(accept shipment)=

Question

A pharmaceutical company receives large shipments of ibuprofen tablets and uses an acceptance sampling plan. This plan randomly selects and tests 20 tablets, then accepts the whole batch if there is at most one that doesn't meet the required specifications. What is the probability that this whole shipment will be accepted if a particular shipment of thousands of ibuprofen tablets actually has a 6% rate of defects?

(Report answer as a decimal value accurate to four decimal places.)

P(accept shipment) = 

Answer 1

The process of randomly selecting 20 ibuprofen tablets and recording the number of tablets out of specification can be thought of as a binomial experiment.

A binomial experiment consists of n identical trials with a probability of "success" p on each trial. The probability of k successes in n trials can be calculated using a computer program, calculator, or by hand using the Binomial Formula: P(x = k) = n!•p^k•q^n-k÷[k!(n-k)!].

Based on the information given, we have

n = 20 trials (ibuprofen tablets)

p = 0.94 (probability of a tablet meeting specification)

q = 0.06 (probability of tablet not meeting specification)

The question tells us that the batch will be accepted if there is at most 1 tablet out of specification. This means the probability of a batch being accepted is the is probability of 19 out of 20 tablets being within specification plus the probability of 20 out of 20 tablets being within specification.

P(accept) = P(x = 19) + P(x = 20), where x is the number of tablets in specification. Let's calculate the binomial probabilities of these two outcomes:

P(x = 19) = 20!•(0.94)¹⁹(0.06)¹÷(19!•1!) = 0.3703

P(x = 20) = 20!•(0.94)²⁰(0.06)⁰÷20! = 0.2901

Therefore, P(accept) = 0.3703 + 0.2901 = 0.6604