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I have been given a tutorial question and I know how to do it.

Suppose that $\Omega = \{1, 2, 3, ...\}$ is a countable sample space. Show that the function $P(k) = \frac{3}{4^k}$ for $k = 1, 2, 3, ...$ is a probability function on the sample space $\Omega$.

I know that it is a geometric series but I do not understand what it is trying to do. Is it because I do not fully understand a probability function or measure theory (something I came across while trying to better understand my Probability and Statistics course). If anyone can give me the intuition or real world application behind it, that would be great.

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  • $\begingroup$ Suppose an apple tree drops each year a certain number of apples, but at least 1, and the probability that it drops $k$ apples is $P(k)$. This only makes sense if the sum over the probabilities of all possible events (It drops 1 apple, 2 apples, ...) is 1. $\endgroup$
    – S. M. Roch
    Commented Nov 19, 2017 at 11:52

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Because $\Omega$ is a countable set, you must check that $\sum_{k\in\Omega}P(k)=1$.

This follows from the geometric series

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