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# Probability

The proportion of times something would occur over the long run.

## Definitions

### Sample Space

With Replacement

Without Replacement

### Simple Event

Sum of all Simple Events = 1

If equally likely then probability = 1/k

## Events

### Conditional Probablity

P(B|A) = probability of B given A

## Rules for Finding Probabilities

### Rule 1 (Not the event)

Probability an event does not occur, P(A)+P(A^c)=1

Probability that either of two events happens, Rule 2a (general), P(A or B)=P(A)+P(B)-P(A and B), Rule 2b (mutually exclusive events), P(A or B)=P(A)+P(B)

### Rule 3 (Multiplication)

Probability that two or more events occur together, Rule 3a (general), P(A and B)=P(A)PB|A)=P(B)PA|B), Rule 3b (independent events), PA and B)=P(A)P(B), Rule 3b (extension), P(A1 and A2...and An)=P(A1)P(A2)...(PAn)

### Rule 4 (Conditional Probability)

Determining a conditional probability, P(B|A)=P(A andB)/P(A)

## Strategies

### Bayes Rule

P(A|B)=P(A and B)/PB|A)P(A)+P(B|A^c)P(A^c)