A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 38 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If one shipment of 5000 aspirin tablets actually has a 3% rate of defects, what is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?

Since the combined probability is less than 1,  the likelihood of defective tablets is very low hence the shipment is likely to be accepted.

Explanation:

Probability is defined as the likelihood of an event occurring or happening.

From the question since only 38 tablets will be randomly sampled out of a shipment of 5000 tablets, our sample space n will be 38

n = 38

Rate of defective tablets = 3% (0.03)

Rate of non-defective tablets = 97% (0.97)

Lets call the Probability of having no defective tablets within the sample space of 38, P₀

                                           P₀   = (non-defective rate)ⁿ

                                                 =(0.97)³⁸

                                                 = 0.3142

Lets call the Probability of having 1 defective tablet out of the sample space of 38,  P₁. therefore

                                              P₁   = 38 x 0.03 x (0.97)³⁷ (which is the product of the sample space by the rate of defect by the no of possible occurrences)

                                               P₁  = 0.3693

Combined probability  (P₀ + P₁), which is the the probability that there is only 1 defective tablet or no defective tablet in the shipment

                                                    = (0.3693 + 0.3142)

                                                     = 0.6835

Since the combined probability is less than 1,  the likelihood of defective tablets is very low hence the shipment is likely to be accepted.

Similar shipments with combined probability less than 1, will also likely be accepted.