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# Probability

a number between 0 and 1 that is assigned to a possible outcome of a random circumstance

## Types of Events

### simple event

one outcome in a sample space, or the possible outcome of a random circumstance

### mutually exclusive events

two events which do not contain any of the same simple events, or outcomes

disjoint event

### independent events

knowing that one event will occur (or has occurred) does not change the probability that the other occurs

### dependent events

knowing that one event will occur (or has occurred) changes the probability that the other occurs

opposite events

## Interpretations of Probability

### relative frequency

proportion of times a specific outcome would occur over the long run

### personal probability

the degree to which a given individual believes that an event will happen

subjective probability

### coherent probabilities

the probability of one event doesn't contradict the probability of another

## Key Definitions

### random circumstance

one in which the outcome is unpredictable

### sample space

the collection of unique, nonoverlapping possible outcomes of a random circumstance

### conditional probability of the event B, given that the event A occurs: P(A|B)

the long-run relative frequency with which event B occurs when circumstances are such that A also occurs

### tree diagram

a schematic representation of the sequence of events and their probabilities, including conditional probabilities based on previous events for events that happen sequentially

### event

any collection of one or more possible outcomes

opposite

## Probability Rules

### Rule 1: "not the event"

To find the probability of A^C, the complement of A, use P(A^C) = 1-P(A)

### Rule 2: addition rule for "either or"

To find the probability that either A or B or both happen

Rule 2a: general

Rule 2b: mutually exclusive events

### Rule 3: multiplication rule for "and"

To find the probability that two events A and B both occur simultaneously or in a sequence

Rule 3a: general

Rule 3b: independent events

Extension of Rule 3b: more than two independent events

### Rule 4: conditional probability

To find the probability that B occurs given that A has occurred: P(B|A) = P(A and B)/P(A)

### Bayes Rule: finding P(A|B) using P(B|A)

P(A|B) = P(A and B) / [P(B|A)P(A) + P(B|A^C)P(A^C)]

### Sample with replacement

a sample in which individuals are returned to the eligible pool for each selection

### Sample without replacement

a sample in which individuals are not eligible for subsequent selection