Sampling With Replacement
Sampling is called with replacement when a unit selected at random from the population is returned to the population and then a second element is selected at random. Whenever a unit is selected, the population contains all the same units, so a unit may be selected more than once. There is no change at all in the size of the population at any stage. We can assume that a sample of any size can be selected from the given population of any size.
This is only a theoretical concept, and in practical situations the sample is not selected by using this selection method. Suppose a population size and sample size , and sampling is done with replacement. Out of elements, the first element can be selected in ways. The selected unit is returned to the main lot and now the second unit can also be selected in ways.
Thus in total there are samples or pairs which are possible. Suppose a container contains good bulbs denoted by and and defective bulbs denoted by and . If any two bulbs are selected with replacement, there are possible samples, as listed in the table below:




































The number of samples is given by . The selected sample will be any one of the possible samples. Each sample has an equal probability of selection. A sample selected in this manner is called a simple random sample.