Events & probabilities
The intersection of two events $$A$$ and $$B$$ is $$A\cap B$$.
The union of $$A$$ and $$B$$ is $$A\cup B$$.
$$A$$ and $$B$$ are mutually exclusive if they cannot both occur, denoted $$A\cap B=\emptyset$$, where $$\emptyset$$ is called the null event.
For an event $$A$$, $$0\leq P(A )\leq 1$$.
For two events $$A$$ and $$B$$ \[P(A\cup B) = P(A) + P(B) - P(A\cap B).\] If $$A$$ and $$B$$ are mutually exclusive then \[P(A\cup B) = P(A) + P(B).\]Equally likely outcomes
If a complete set of $$n$$ elementary outcomes are all equally likely to occur, then the probability of each elementary outcome is $$\displaystyle\frac{1}{n}$$. If an event $$A$$ consists of $$m$$ of these $$n$$ elements, then $$\displaystyle P(A) = \frac{m}{n}$$.
Independent events
$$A$$, $$B$$ are independent if and only if $$P(A\cap B)=P(A)P(B)$$.
Conditional Probability of $$A$$ given $$B$$: \[P(A\,|\,B)=\frac{P(A\cap B)}{P(B)},~~~~~ P(B) \neq 0.\]
Bayes' Theorem: $$\displaystyle P(B\,|\,A)=\frac{P(A\,|\,B)P(B)}{P(A)}$$
Theorem of Total Probability
The $$k$$ events $$B_1,B_2,\ldots,B_k$$ form a partition of the sample space $$S$$ if $$ B_1\cup B_2\cup\ldots\cup B_k=S$$ and no two of the $$B_i$$'s can occur together. Then $$P(A)=\sum{P(A\,|\,B_i)P(B_i)}$$. In this case Bayes Theorem generalizes to \[P(B_i\,|\,A)=\frac{P(A\,|\,B_i)P(B_i)}{\sum_{j}{P(A\,|\,B_j)P(B_j)}},~(i= 1, 2, \ldots, k).\] If $$B^{\prime}$$ is the complement of the event $$B$$, $$P(B^{\prime})=1 - P(B)$$ and $$P(A)=P(A\,|\,B)P(B)+ P(A\,|\,B^{\prime})P(B^{\prime})$$ is a special case of the theorem of total probability. Also, the complement of the event $$B$$ is commonly denoted $$\bar{B}$$.