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Sample space: the possible outcomes of the experiment. Conceptually, it is a set. It can be discrete, continuous, finite, infinite and so on.

The elements of the sample space follow certain properties: The elements should be mutually exclusive, collectively exhaustive and at the right granularity.

Probability Law: assigns probabilities to outcomes or to collections of outcomes. It tells us whether one outcome is much more likely than some other outcome.

Probability Axioms and Properties

Event: a subset of the sample space. Probability is assigned to events. Axioms: * Nonnegativity: $(P(A)\geq 0)$. (Any event cannot have negative probability) * Normalization: $(P(\Omega)=1)$. (The sum of the probability of all events equals to 1) * Finite Additivity: If $A\cap B=\emptyset$ (These two events disjoint to each other), then $P(A\cup B)=P(A)+P(B)$. * Countable Additivity Axiom: If $A1, A2, \cdots $ is an infinite sequence of disjoint events, then $P(A1\cup A2\cup\cdots)=P(A1)+P(A2)+\cdots$. (Probability of disjoint events equals to the sum of their probabilities)

With these axioms, we can infer the following properties:

Properties: * $P(A)\leq 1$. * $P(\emptyset)=0$. * $P(A)+P(A^c)=1$. * For $k$ disjoint events, $P({s1, s2,\cdots,sk})=P(s1)+\cdots+P(s_k)$. * If $A\subset B$, then $P(A)\leq P(B)$. * $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. * $P(A\cup B)\leq P(A)+P(B)$. This property is called the Union Bound. * $P(A\cup B\cup C)=P(A)+P(A^c\cap B)+P(A^c\cap B^c\cap C)$.

Uniform Law

Discrete: Assume $\Omega$ consists of $n$ equally likely elements, and assume $A$ consists of $k$ elements. Then $P(A)=\frac{k}{n}$. Continuous: Probability = Area

Probability Calculation Steps * Specify the sample space. * Specify a probability law. * Identify an event of interest. * Calculate the probability of the event of interest.

Bonferroni's Inequality * For any two events $A1$ and $A2$, we have $P(A1\cap A2)\geq P(A1)+P(A2)-1$. * Generally, we have $P(A1\cap A2\cap\cdots An)\geq P(A1)+P(A2)+\cdots+P(An)–(n-1)$.