The assertion 'everybody participating in the lottery can be a winner' is ambiguous. It may say that it is possible that all participants win, and it may say that for all participants it is possible to win.

My question is: 'what are the logical formulas to explicate this ambiguity'? I am interested in formulas in first order predicate logic and modal logic. Who can help?

Hi Rebel,
I am not interested in symbolic logic formulas, but I wish to point out the following to you.
1) You are perfectly able to formulate the problem to everyone's comprehension through ordinary language, i.e. plain English.
2) The distinction of interest to you here is that between a distributive proposition (which applies the predicate to every instance of the subject, i.e. to all of them taken each one singly, i.e. to every one) and a 'collectional' proposition (which tells us that all instances of the subject will get the predicate, not only each one singly, but also at the same time).
See this topic in Future Logic, chapt. 19.4

Something about you (optional) writer in logic, philosophy, spirituality