I think you forgot to mention the condition 'given that you already have a ticket', and whatever justifications are required to assume that two more winning tic

I think you forgot to mention the condition 'given that you already have a ticket', and whatever justifications are required to assume that two more winning tickets will be present (if each ticket has independent odds of being a winner then you end up with a distribution of other tickets yeah?). Otherwise your premise doesn't quite lead to your conclusion.