Question

The probability of selecting a sample of size 'n' out of a population of size N by simple random sampling with replacement is:

Answer 1

When n units are selected with Simple Random Sampling With Replacement(SRSWR), the total number of possible samples are Nⁿ . The
Probability of drawing a sample is 1/Nⁿ
Alternatively, let i u be the i^th unit selected in the sample. This unit can be selected in the sample
either at first draw, second draw, …, or n^th draw. At any stage, there are always N units in the
population in case of SRSWR, so the probability of selection of i^th unit at any stage is 1/N for all i = 1,2,…,n.
Then the probability of selection of n units u₁,u₂,......,u_n
P(u₁,u₂,......,u_n )
=P(u₁).P(u₂).......P(u_n)
=
$$\begin{gathered} \frac{1}{N}\frac{1}{N}\frac{1}{N}......\frac{1}{N} \\ =\frac{1}{N^n} \end{gathered}$$