# Is there a law or theorem related to occurrence of an event with highest probability in a population with infinite size?

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.

• Yes, I mean the categorical (also called Multinoulli) distribution which can have more than 2 possible outcomes. A Multinoulli can easily be viewed as though it were Bernoulli, by defining a random variable that is 1 if an outcome happens, and 0 if that outcome doesn't happen. Then at every trial you have an $X_i$ that is either 0 or 1, and the observed proportion of it is $\sum_{i=1}^n X_i / n$. This is a sample mean, and each of the 3 keys has this kind of sample mean by viewing it as a Bernoulli, hence the LLN applies to all 3 of them. Aug 24, 2023 at 0:13
• Based on this analogy, we can say that each room is a separate variable. For example, the red room corresponds to $X_1$, $X_1=0$ corresponds to key not appearing in the red room, and $X_1=1$ corresponds to key appearing in the red room. When $n\rightarrow\infty$ we have $\sum_{i=1}^{n}X_i/n=0.4$. Is that true? Aug 24, 2023 at 0:21