<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/555aab25-9bfe-4d4a-bc55-a2fbadd4487a/Random_Processes.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/555aab25-9bfe-4d4a-bc55-a2fbadd4487a/Random_Processes.png" width="40px" /> Random processes: are those which cannot be predicted ahead of time
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<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/b86b9329-dc84-484a-b8f4-0c99a9280da4/Outcome.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/b86b9329-dc84-484a-b8f4-0c99a9280da4/Outcome.png" width="40px" /> Outcome: is the result of a random process
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<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/c5301512-e133-43f6-9c88-49dc60467406/Sample_space.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/c5301512-e133-43f6-9c88-49dc60467406/Sample_space.png" width="40px" /> Sample space: the range of all possible outcomes
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In the case where there can only be $2$ outcomes $A$ and $B$:
$$ P(A)=\frac{\overbrace{N_A}^{\text{Nb of outcome A}}}{\underbrace{N_A+N_B}_{\text{Tot. Nb. of outcomes}}} $$
<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/164cdeac-a475-4401-9110-c1a2173d3778/Independent.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/164cdeac-a475-4401-9110-c1a2173d3778/Independent.png" width="40px" /> Independent: If the probability of A does not change if we only consider the sample space filled by B
if A = True : B = '?'
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<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/a2753650-6383-43b8-8641-02cf55ed700f/Mutually_exclusive.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/a2753650-6383-43b8-8641-02cf55ed700f/Mutually_exclusive.png" width="40px" /> Mutually exclusive: If two sets of outcomes do not overlap in sample space, they are mutually exclusive
if A = True : B = False
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<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/b5626bb2-fb66-4e58-b3c8-52390b2535f2/dependent.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/b5626bb2-fb66-4e58-b3c8-52390b2535f2/dependent.png" width="40px" /> Dependent: the outcome of the first event influences the outcome of the second event
A = f(B)
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$$ \begin{aligned} \textbf{Axiom 1:}& \quad0\le P(A)\le1 \\ \textbf{Axiom 2:}& \quad P(A\cup \bar A)=1 \\ \textbf{Axiom 3:}& \quad \text{for mutually exclusive outcomes:} \\ &\qquad P(A_1\cup A_2 \cup \cdots \cup A_N)=\sum^N_{i=1}P(A_i) \end{aligned}
$$
$$ P(A\cup B)=P(A)+P(B)-P(A\cap B) $$
$$ p(A\cap B)=P(A)\times P(B) $$
$$ P(A_1\cup A_2\cup\cdots\cup A_N)=1-\prod^N_{i=1}P(\bar A_i) $$