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Let \((\Omega, \mathcal{F}, P)\) be a probability space and let \(A\) and \(B\) be events in the event space \(\mathcal{F}\) (\(A, B \in \mathcal{F}\)). The probability function, \(P\), must satisfy the following axioms which we accept as truth.
That is, the resulting probability of an event occurring must be a number between \(0\) and \(1\).
That is, the probability that at least one of the elementary events in the sample space will occur is \(1\) [1]. Logically, this means that an outcome from any trial of the experiment will be in the sample space, \(\Omega\) meaning that all outcomes must be from the sample space, \(\Omega\). Intuitively, this also means that the sum of the probabilities for each element in the sample space is equal to \(1\).
where \(A\) and \(B\) are mutually exclusive events (they cannot both occur at the same time meaning the sets are disjoint: an example is coin tossing. A trial can result in heads or tails but not both). In words, this means that the probability of \(A\) or \(B\) occurring is the sum between the probability that \(A\) will occur and the probability that \(B\) will occur. This generalizes beyond just two events [2]. Specifically, for any sequence of mutually exclusive events \((E_1, E_2,...)\):
\[P\left(\bigcup_{i=1}^{\infty} E_{i}\right)=\sum_{i=1}^{\infty} P\left(E_{i}\right).\]Citations:
[1] - https://en.wikipedia.org/wiki/Probability_space
[2] - Sheldon M Ross. A first course in probability. Pearson, 2014.
Also: CS 109 Course Reader