# Sampling probability

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{{short description|Theory relating to sampling from finite populations}}
In [statistics](/source/statistics), in the theory relating to [sampling](/source/Sampling_(statistics)) from finite [populations](/source/Statistical_population), the '''sampling probability''' (also known as '''inclusion probability''') of an [element](/source/Element_(statistics)) or member of the population, is its [probability](/source/probability) of becoming part of the sample during the drawing of a single sample.<ref>{{cite book |last=Dodge |first=Y. |year=2003 |title=The Oxford Dictionary of Statistical Terms |publisher=OUP |isbn=0-19-850994-4 }}</ref> For example, in [simple random sampling](/source/simple_random_sample) the probability of a particular unit <math>i</math> to be selected into the sample is
:<math>p_{i} = \frac{\binom{N-1}{n-1}}{\binom{N}{n}} = \frac{n}{N}</math>
where <math>n</math> is the sample size and <math>N</math> is the population size.<ref>{{cite book |first1=Adrian |last1=Baddeley |author-link=Adrian Baddeley |first2=Eva B. |last2=Vedel Jensen | author2-link = Eva Vedel Jensen |title=Stereology for Statisticians |year=2004 |page=334 |url=https://books.google.com/books?id=il0fXb_GSowC&pg=PA334 }}</ref>

Each element of the population may have a different probability of being included in the sample.  The inclusion probability is also termed the "first-order inclusion probability" to distinguish it from the "second-order inclusion probability", i.e. the probability of including a pair of elements. Generally, the first-order inclusion probability of the ''i''th element of the population is denoted by the symbol π<sub>''i''</sub> and the second-order inclusion probability that a pair consisting of the ''i''th and ''j''th element of the population that is sampled is included in a sample during the drawing of a single sample is denoted by π<sub>''ij''</sub>.<ref>{{cite book |last1=Sarndal |last2=Swenson |last3=Wretman |year=1992 |title=Model Assisted Survey Sampling |publisher=Springer-Verlag |isbn=0-387-40620-4 }}</ref>

==See also==
*[Sampling bias](/source/Sampling_bias)
*[Sampling design](/source/Sampling_design)
*[Sampling frame](/source/Sampling_frame)

==References==
{{Reflist}}

==Further reading==
*{{cite book |first=M. E. |last=Thompson |authorlink=Mary E. Thompson |title=Theory of Sample Surveys |year=1997 |chapter=The mathematics of probability sampling designs |pages=9–48 |publisher=Taylor & Francis |isbn=0-412-31780-X }}

Category:Sampling (statistics)

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