12th International Statistics Days Conference, İzmir, Turkey, 13 - 16 October 2022, pp.44
Let there be N items partitioned into a number of groups. Group i has size n_i and weight w_i. Consider the sampling procedure of choosing n items as follows: in each round one of the groups is randomly selected to include in the sample all items therein, and the probability of selecting a group is equal to its weight divided by the yet unselected groups' total weight. The special case where groups consist of one item only is referred to as "weighted random sampling without replacement with defined weights." We abbreviate this as WRS, and the general case, which we shall call "weighted random sampling of groups without replacement with defined weights," as WRSG. Let p_i denote the probability that group i is included in the sample. Even for WRS there exists no efficient method in the literature to calculate p_i exactly. This presentation's subject matter is an investigation of these inclusion probabilities. Our motivation comes from a real-life problem related to hajj draws, the so-called "katsayılı kura sistemi" in Turkish, conducted every year by some countries including Bosnia and Herzegovina and Belgium. To summarize, we derive theoretical lower and upper bounds on inclusion probabilities in terms of item weights. We use these bounds as well as simulation to estimate applicants' chances in Turkey's 2020 hajj draw. Our results rely on a conjecture for which we provide a supportive example. It turns out that one who participates in the draws for the first time has a chance in between %0.12 and %0.13; similar bounds for one who participates for the eleventh time are %13.22 and %14.16.