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# 1 Introduction to Probability

### Definitions of Probability:

1. Classical Definition: $$P(A)=\frac{\text{# of ways event A can occur}}{\text{# of outcomes in S}}$$
2. Assumption: All outcomes are equally likely.

3. Relative Frequency Definition: $$P(A)=\frac{\text{# of ways event A occured}}{\text{# of repetitions of experiment}}$$
4. Limitation: Can never repeat an experiment indefinitely or even a long series of repetitions.

5. Subjective Probability Definition: The probability of an event is a measure of how sure the person making the statement is that the event would occur.
6. Limitation: No rational basis for agreement on a right answer.

# 2 Mathematical Probability Model

Definition: A sample space, denoted by $S$, refers to the set of all possible outcomes of an experiment.
e.g. $S=\{\text{"heads, "tails"}\}$.
A sample space can be classified as:
• Discrete
• Takes only integer values
• Things we count
• Can be further classified as:
• Finite: e.g. rolling a die twice
• Countably Infinite: e.g. tossing a coin till a head appears
• Continuous
• Takes any real values
• Things we measure
Definition: An event is a subset of the sample space. Probabilities are defined on events.
Events can be classified as:
• Simple Event: An event that consists of 1 element
• Compound Event: An event that has more than one element
Definition: An probability function, P, is a function that satisfies the following 2 axioms:
1. P(S)=1; P($\phi$)=0
2. $P(\cup_{i=1}^\infty A_i)=\sum_{i=1}^\infty P(A_i)$, $A_i\cap A_j=\phi$ ($A_i$ and $A_j$ are "mutually exclusive" or "disjoint")
If 2 events are disjoint, then $P(A_i\cup A_j)=P(A_i)+P(A_j)$
Definition: Two events are exhaustive if at least one of them has to occur. i.e. $A\cup B=S$