Membership in random ratio sets

Let A be a random set constructed by picking independently each element of {1,…,n} with probability α∈(0,1). We give formula for the that rational number q belongs to ratio A/A≔{a/b:a,b∈A}. This generalizes previous result Cilleruelo and Guijarro-Ordóñez. Moreover, we make some considerations about formulas event ⋁i=1k(qi∈A/A), where q1,…,qk are numbers, showing they related study connected components certain graphs. In particular, qe∈A/A e∈E, E is finite or cofinite positive integers 1∈E.