Apharmaceutical company receives large shipments of aspirin tablets. the acceptance sampling plan is to randomly select and test 2626 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. if a particular shipment of thousands of aspirin tablets actually has a 44% rate of defects, what is the probability that this whole shipment will be accepted?

For a really large shipment, and a relatively high "success" rate (i.e. selecting the defectives), we may assume that sampling without replacement does not change the probability, hence binomial distribution may be used, with which

where and
n=26, p=0.44

In the given case, n=26, we need
P(0)+P(1)


=0.000006080 
with a 4% error compared with the exact answer (for a population of 5000), or 0.42% error if the population is 50,000, as follows

Exact solution by hypergeometric distribution taking into account of selection without replacement, assuming a population of only 5000 
P(x)=C(A,a)C(B,b)/(C(A+B,a+b)
a={0,1}
b={26,25}
A={2200,2199}
B={2800,2800}

P(0)+P(1)
=C(2200,0)C(2800,26)/C(5000,26)+C(2200,1)C(2800,25)/C(5000,26)
=0.0000002695+0.0000055557
=0.00058252

Similarly, for a 50000 population,
=C(2200,0)C(2800,26)/C(5000,26)+C(2200,1)C(2800,25)/C(5000,26)
=0.0000060539

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