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Probability
by Chelsea Gray
# Probability

## Random Circumstances

### Random circumstance: one in which the
outcome is unpredictable. In many cases the
outcome is not determined until we observe
it.

### Probability: a number between 0 and 1 that
is assigned to a possible outcome of a
random circumstance.

## Interpretations of Probability

### Relative frequency: In situations that we can imagine
repeating many times, we define the probability of a specific
out come as the proportion of times it would occur over the
long run.

### Personal probability: the degree to which a
given individual believes that the event will
happen

### Subjective probability: The degree of
belief may be different for each
individual

### Coherent: your personal probability of one
event doesn't contradict your personal
probability of another

## Probability Definitions and
Relationships

### Sample space: the collection of unique,
non-overlapping possible outcomes of a
random circumstance.

### Complementary event: the two events do not
contain any of the same simple events and
together they cover the entire sample space.

### Mutually exclusive/disjoint: the events do not
contain any of the same simple events
(outcomes).

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

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

### Conditional probability: of the event B, given that the event
A occurs, is the long-run relative frequency with which
event B occurs when circumstances are such that A also
occurs.

## Basic Rules for Finding
Probabilities

### Rule 1 ( "not the event"): to find the probability
of A^c, the complement of A, use P(A^c) = 1 -
P(A). P(A^c) is the probability that event A won't
occur

### Rule 2 (addition rule for "either/or"): to find
the probability that either A or B or both
happen.

### Rule 3 (multiplication rule for "and"): to find
the probability that two events A and B both
occur simultaneously or in a sequence.

### Rule 4 (conditional probability):
an algebraic restatement of rule
3a

### Sample with replacement: individuals are
returned to the eligible pool for each
selection.

### Sample without replacement:
individuals are not eligible for
subsequent selection.

## Strategies for Finding
Complicated Probabilities

### Bayes Rule: P(A|B) = P(A and
B)/P(B|A)P(A) + P(B|A^c)P(A^c)

### Tree diagram: a schematic representation of the sequence of
events and their probabilities, including conditional
probabilities based on previous events that happen
sequentially.

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Simple Event: one outcome in the sample space. Simple event is a possible outcome of a random circumstance., Event: a collection of one or more simple events in the sample space. Often written using capital letters.

Rule 2a (general): P(A or B) = P(A and B)

Rule 2b (for mutually exclusive events): if A and B are mutually exclusive events, P(A or B) = P(A) + P(B)

Rule 3a (general): P(A and B) = P(A)P(B|A) = P(B)P(A|B)

Rule 3b (for independent events): if A and B are independent events, P(A and B) = P(A)P(B)

P(B|A) = P(A and B)/P(A)