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 $$N = 5$$ and sample size $$n = 2$$, and sampling is done with replacement. Out of $$5$$ elements, the first element can be selected in $$5$$ ways. The selected unit is returned to the main lot and now the second unit can also be selected in $$5$$ ways.
Thus in total there are $$5 \times 5 = 25$$ samples or pairs which are possible. Suppose a container contains $$3$$ good bulbs denoted by $${G_1},{G_2}$$ and $${G_3}$$ and $$2$$ defective bulbs denoted by $${D_1}$$ and $${D_2}$$. If any two bulbs are selected with replacement, there are $$25$$ possible samples, as listed in the table below:

$${G_1}$$

$${G_2}$$

$${G_3}$$

$${D_1}$$

$${D_2}$$

$${G_1}$$

$${G_1}{G_1}$$

$${G_1}{G_2}$$

$${G_1}{G_3}$$

$${G_1}{D_1}$$

$${G_1}{D_2}$$

$${G_2}$$

$${G_2}{G_1}$$

$${G_2}{G_2}$$

$${G_2}{G_3}$$

$${G_2}{D_1}$$

$${G_2}{D_2}$$

$${G_3}$$

$${G_3}{G_1}$$

$${G_3}{G_2}$$

$${G_3}{G_3}$$

$${G_3}{D_1}$$

$${G_3}{D_2}$$

$${D_1}$$

$${D_1}{G_1}$$

$${D_1}{G_2}$$

$${D_1}{G_3}$$

$${D_1}{D_1}$$

$${D_1}{D_2}$$

$${D_2}$$

$${D_2}{G_1}$$

$${D_2}{G_2}$$

$${D_2}{G_3}$$

$${D_2}{D_1}$$

$${D_2}{D_2}$$

The number of samples is given by $${N^n} = {5^2} = 25$$. The selected sample will be any one of the $$25$$ possible samples. Each sample has an equal probability $$1/25$$ of selection. A sample selected in this manner is called a simple random sample.
M A Hafez
November 3 @ 11:58 am
What is the limitation of this type of sampling? In which kind of situation(s) sampling with replacement is recommended?
M A hafez
November 3 @ 12:00 pm
Initial explanation is excellent.