Assume, we have a key that appears in either of the three rooms randomly (red room, blue room, and green room). We have the following probability distribution:
P(key appears in red room) = 0.35
P(key appears in green room) = 0.40
P(key appears in blue room) = 0.25
It is intuitive to say that when the key randomly appears in a room, the number of times that the key appears in the green room is more than the times the key appears in the other rooms with a guarantee when the size of the population of random occurrences is large enough. This sounds like the law of large numbers (LLN) but it is not quite like that since LLN talks about an average of a sample. Is there any theorem or law that can back up this claim?
Note: there is only one key that randomly appears in either of the rooms at each time instance.