Simple Random Sampling

Simple random sampling is a simple and widely used method of sampling in which the samples with equal probability of being selected at each draw, are selected unit by unit. It is a technique of drawing a sample in such a way that each population unit has an equal and independent chance of being included in the sample. There are two ways of drawing a sample through simple random sampling:

1. Simple random sampling with replacement: In this sampling, an item previously drawn is replaced before the next draw in which the same unit may be included more than once in the sample.
2. Simple random sampling without replacement: In this method, an item previously drawn is not replaced before the next draw in which all units in the sample are distinct.

1. Simple random sampling with replacement (SRSWR): A simple random sampling with replacement (SRSWR) of size n from a population of N units can be imagined as an act of drawing independent samples of size 1. A unit is randomly selected from the population to be the first sampled unit, with probability 1/N. Then, the sampled unit is replaced in the population, and a second unit is randomly selected with probability 1/N. This procedure is repeated until the sample has n units, which may include duplicates from the population. The number of possible sample having sample size n from the population size N is Nⁿ. Hence, the probability of one sample = 1/Nⁿ.
   However, in finite population sampling, sampling the same person twice provides no additional information. We usually prefer to sample without replacement, so that the sample contains no duplicates.
2. Simple random sampling without replacement (SRSWOR): A simple random sample without replacement (SRSWOR) of size n is selected so that every possible subset of n distinct units in the population has the same probability of being selected as the sample. One unit is randomly selected from the population to be first sampled unit with probability 1/N. The sampled unit is not replaced and the second unit is randomly selected with probability 1/(N-1). Similarly, the second sampled unit is not replaced and third unit is randomly selected with probability 1/(N-2) and so on until the sample has n units obtained from the population. There are ^NC_n possible samples of size n from population size N and each is equally likely. So, the probability of selecting any individual sample of n units is 1/^NC_n.