In your case ,the concept of Mutually exclusive and independent events seems more accurate as we have multiple variables and different outcomes .Lets start by considering definition of Mutually exclusive events
Two events are said to be independent if the result of the second event is not affected by the result of the first event.
If A and B are independent events, the probability of both events occurring is the product of the probabilities of the individual events
Hence,If A and B are independent events,
P(A and B) = P(A) P(B)
This does satisfy your mathematical notion of Winning in Gambling in long run is possible but disagree with the fact as luck is to be considered as a positive entity in the process.
I dont know, because we dont know how luck works or what luck is.
It arises out of the quantum level but we cannot know how it arises or what is the nature of it. It all comes down to determinism vs free will.
And that will remain a mistery.