why does the probability function must add up to 1? Let $Y$ be a discrete random variable with the probability function $p(y)$. Then the expected value of $Y$, is define to be $\Bbb{E}(Y) = \sum_y p(y)$
(a) Briefly explain why the sum of all $y$, $p(y) = 1$.
The way I can think about this is problem is by the following : if I have a sample space that has $y = 0,1,2,.. n$, number of events and then corresponging probability of each event must add up to 1 because $1 = p(y_0) + p(y_1)+ p(y_2) + \cdots + p(y_n)$.
Is this correct? or is there a better way to explain it.


If we take the axiomatic definition of probability, then we have
$$P(\Omega) = 1$$
where $\Omega$ denotes the whole space of events i.e. any possible outcome is in $\Omega$.
When you sum through all the values of $y $ you are summing up through all the possible outcomes. That means you are summing through all events in $\Omega $ and as we know (because of our definition of probability), $P(\Omega) = 1$.
