# 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.