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**Sampling from a Population with Two Categories of Elements**

Consider a lot (population) of items in which each item can be classified as either defective or nondefective. Suppose that a lot consists of 15 items, of which 5 are defective and 10 are nondefective. Do the conditions for Bernoulli trials apply when sampling (1) with replacement and (2) without replacement?- 1
**Sampling with replacement**

An item is drawn at random (i.e., in a manner that all items in the lot are equally likely to be selected). The quality of the item is recorded and it is returned to the lot before the next drawing. The conditions for Bernoulli trials are satisfied. If the occurrence of a defective is labeled*S*, we have*P*(*S*)=5/15.- 2
**Sampling without replacement**

In situation (1), suppose that 3 items are drawn one at a time but without replacement. Then the condition concerning the independence of trials is violated. For the first drawing,*P*(*S*)=5/15. If the first draw produces*S*, the lot then consists of 14 items, 4 of which are defective. Given this information about the result of the first draw, the conditional probability of obtaining an*S*on the second draw is then , which establishes the lack of independence.

This violation of the condition of independence loses its thrust when the population is vast and only a small fraction of it is sampled. Consider sampling 3 items without replacement from a lot of 1500 items, 500 of which are defective . With

*S*_{1}denoting the occurrence of an*S*in the first draw and*S*_{2}that in the second, we have and .For most practicl purposes, the latter fraction can be approximated by 5/15. Strictly speaking, there has been a violation of the independence of trials, but it is to such a negligible extent that the model of Bernoulli trials can be assumed as a good approximation.

If elements are smpled from a dichotomous population at random and with replacement, the conditions for Bernoulli trials are satisfied. When the sampling is made without replacement, the condition of the independence of trials is violated. However, if the population is large and only a small fraction of it (less than 10%, as a rule of thumb) is sampled, the effect of this violation is negligible and the model of the Bernoulli trials can be taken as a good approximation.