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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.

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1 Answer 1

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Under the hood there is actually a mean here. A proportion, namely, is a sum of 0/1 random variables divided by their sample size. This means that the LLN is applicable to proportions as well, and in the limit the proportion of keys in the green room will converge to 0.40.

Using the LLN, a broader result known as the Glivenko-Cantelli theorem can be proven, which roughly speaking states that the empirical CDF converges uniformly to the theoretical CDF. This means that in the limit, the proportions of all 3 keys will correspond to their theoretical probabilities, since the observed CDF converges to the categorical distribution you give. This also implies that the probability of the key in the green room is larger than that of the other rooms.

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  • $\begingroup$ Thanks for your answer! Just for clarification, in the first part, you mean that we have a variable that can get three values: 0 (red room), 1 (green room), and 2 (blue room)? Could you elaborate on the first part, please with an example? $\endgroup$
    – Breeze
    Aug 24, 2023 at 0:03
  • $\begingroup$ I also edited the post for some clarifications. $\endgroup$
    – Breeze
    Aug 24, 2023 at 0:07
  • $\begingroup$ 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. $\endgroup$ Aug 24, 2023 at 0:13
  • $\begingroup$ Thanks for your reply! We have only one key actually. I added a note at the end of the post to clarify that. $\endgroup$
    – Breeze
    Aug 24, 2023 at 0:15
  • $\begingroup$ 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? $\endgroup$
    – Breeze
    Aug 24, 2023 at 0:21

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