I'm looking for one specific sampling method that decides about inclusion probability of each item regardless of existence of other elements. As an example given 0.5 as the inclusion probability, it toss a coin for each element to decide about its inclusion.
This is different from fixed size sampling methods like sampling with replacement and without replacement as in this method only inclusion probability is provided and not the sample size.
Now I need more information about this method like the name of method, the probability of having different sizes for the sample and ...
Thanks.
It would help to know the application you have in mind. In some applications this is called 'random thinning'. You may want to search the Internet for that phrase.
For example, suppose you have a Poisson process. You may choose (or only be capable of) selecting or observing only a fraction $p$ of the events at random. In this application, if the parent Poisson process has rate $\lambda$, then the sampled or observed process is again Poisson, but with rate $p\lambda.$ In such a context, random thinning is sometimes called 'filtering'.
In Markov Chain Monte Carlo procedures, the $B$ generated points will typically not be independent. By deterministic thinning (say selecting every tenth point) or random thinning (selecting one tenth of the points at random) one obtains a sample that is essentially iid. The sparseness of the thinning depends on the autocorrelation function (ACF) of $B$ points generated by the MCMC procedure. If random thinning is used the thinned sample size will be binomial with mean $B/10$.
In Monte Carlo integration, certain kinds of 'acceptance sampling' sample at random without a fixed sample size.
In certain kinds of 'cluster sampling' of, say, a human population, one may sample census tracts (clusters) at random and then sample households within tracts at random. One or both of the phases of sampling may be proportional rather than based on a fixed sample size. Similar schemes are used in sampling plants or animals. The main motivation here is that it is more convenient or efficient to sample within clusters than randomly across the entire population.
These are three examples that I have encountered personally. And there are others I have not mentioned. The difficulty is that the terminology seems to change depending on the type of application. Maybe if you google some of the phrases I have put in quotes, you will find exactly what you want.
I hope some of this is helpful. If not, perhaps you can be more specific about your application and the reason for selecting without a fixed sample size. Then perhaps I--or someone else--can give a more targeted answer.