Summary: The procedure of choosing an object with special properties sometimes is carried out in two stages. At the first stage, by simplified treatment, suspicious objects are separated, and then they are studied in more detail, beginning with the most suspicious. An important characteristic of such a procedure is the quantity \(\tau\) equal to the number of objects studied in detail in the second stage. In this paper, the distribution of \(\tau\) is studied, and the results are applied to estimating the quality of two-stage procedures of choosing between several polynomial hypotheses.