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acceptable), then the probability distribution of the various possible
outcomes is given by the expansion of the binomial:

ðpþqÞn ð 11 : 1 Þ

Wheren, the number of sample units examined, is small compared with
the lot size.
Since, in a two-class plan, a sample can only be acceptable or defective
then

pþq¼ 1 ð 11 : 2 Þ

The probability that an event will occurxtimes out ofntests is given by:

PðxÞ¼ðn!=ðn
xÞ!x!Þpxqðn
xÞ ð 11 : 3 Þ

or if we substitute (1
p) forq:

PðxÞ¼ðn!=ðn
xÞ!x!Þpxð 1 
pÞn
x ð 11 : 4 Þ

If we have a sampling plan which does not permit any defective samples
(c¼0), then by puttingx¼0 into Equation (11.4) we obtain an expres-
sion for the probability of obtaining zero defective samples, i.e.the
probability of accepting a lot containing a proportionpdefective samples:

Pacc¼ð 1 
pÞn ð 11 : 5 Þ

It follows that the probability of rejection is:

Prej¼ 1 
ð 1 
pÞn ð 11 : 6 Þ

We can use this equation to determine how effective such relatively
simple sampling schemes are. For example, Table 11.1 shows how the
frequency (probability) of finding a defective sample changes with the
level of defectives in the lot as a whole and with the number of samples
taken. This could apply to aSalmonella-testing scheme where detection
of the organism in a single sample is sufficient for the whole lot to be
rejected. If the incidence of Salmonella is 1% (p¼0.01), a level

Table 11.1 Acceptance and rejection thresholds in a sampling scheme

Incidence of defectivesa(%)

No. of samples tested
10 20 100
Pacc Prej Pacc Prej Pacc Prej
0.01 99.9 0.1 99.8 0.2 99.0 1.0
0.1 99.0 1.0 98.0 2.0 90.0 10.0
1 90.0 10.0 82.0 18.0 37.0 63.0
2 82.0 18.0 67.0 33.0 13.0 87.0
5 60.0 40.0 36.0 64.0 0.6 99.4
10 35.0 65.0 12.0 88.0 0.1 99.9

ae.g.Presence ofSalmonellain a sample or surviving mesophiles in packs of an appertized food

400 Controlling the Microbiological Quality of Foods