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probability

is measured between 0 and 1

experimental probability

What the outcomes did turn out to be in an experiment.

Theoretical Probability

What the outcomes were supposed to be theoretically.

complement of an event

all the possible outcomes in the sample space that are not part of the event

probability of a complement

the sum of the probability of an event and the probability of its complement is 1

frequency table

a data display that shows how often an item appears in a category

Relative Frequency

the ratio of the frequency of a category to the total frequency

probability distribution

shows the probability of each possible outcome

Fundamental Counting Principle

(# of choices)(# of choices) = total outcomes

permutataion

an arrangement of items in which the order of the objects is important

factorial

the result of multiplying a sequence of descending natural numbers; symbolized as n! for any number n

permutation notation

the number of permutations of n items of a set arranged r items at a time is the quotient of n factorial and the factorial of the difference of n and r

combination

a selection of items in which order is not important

combination notation

the number of combinations of n items chosen r at a time is the quotient of n factorial and the product of r factorial and the factorial of the difference of n and r

compound event

an event consisting of two or more simple events (choosing a red sock and a blue sock)

independent events

when one event does not effect the other

dependent events

when one events effects the outcome of another event (without replacement)

probability of A and B

if the A and B are independent events, then the probability of (A and B) is the product of the probability of A and the probability of B

mutually exclusive events

events that cannot happen at the same time

probability of mutually exclusive events

if A and B are mutually exclusive events, then the probability of A and B is zero and the probability of (A or B) is the sum of the probability of A and the probability of B

overlapping events

events with common outcomes

probability of overlapping events

if A and B are overlapping events, then the probability of (A or B) is the difference of (the sum of the probability of A and the probability of B) and the probability of (A and B)

two-way frequency table

displays the frequencies of data in two different categories; also known as a contingency table

conditional probability

probability that an event will occur, given that another event has already occured

expected value

the sum of each outcome's value multiplied by its probability

Calculating Expected Value

If A is an event that includes outcomes A₁, A₂, A₃... and Value(A(n)) is a quantitative value associated with each outcome; the expected value of A is the sum of the products of each outcomes probability and expected value

Experiment

an organized procedure for testing a hypothesis.

sample space

is the set of all possible outcomes

Event

a single outcome or a group of outcomes

Simple Event

one outcome or a single collection of outcomes

Discrete Sample Space

A sample space which is either finite, or countably infinite.

A ∪ B

The set of outcomes in the event A, or B, or both.

A ∩ B

The set of outcomes in the event A and B.

A^c

The set of outcomes that are not in A (A's complement).

∅

The empty event (an event that cannot happen).

mutually exclusive

terms do not overlap

P(S) = 1

P(S) - Probability of the Sample Space

If A ⊂ B...

P(B \ A) = P(B) - P(A)

P(A) - Probability of an Event A

P(A) = P(A₁) + P(A₂) + P(A₃)..., where A₋n is an elementary subset of the event A.

|a|

The number of outcomes in an event A.

List the Sample Space Method

Find the probability of an event by listing the sample space, then dividing the number of elements in the event by the total number of elements in the samples space.

Basic Principle of Counting

If operation 1 can be done in m ways, and operation 2 can be done in n ways, then the combined operation can be done in m * n ways.

Permutations

The number of ordered arrangements of n distinct objects, taken r at a time.

Combinations

The number of unordered combinations of n objects, taken r at a time.

Partitioning Objects into Distinct Groups (where each group is of a given size)

To partition n objects into k distinct groups, with each group containing n₋i objects:

P(A|B) = P(A ∩ B) / P(B)

P(A|B) - Probability of Event B given that an Event A has occurred

P(B|B)

P(B|B) = 1

P(A^c|B)

P(A^c|B) = 1 - P(A|B)

P(C ∪ D|B)

P(C ∪ D|B) = P(C|B) + P(D|B)

Partitions

Multiple events form a partition of the sample space if they are pairwise mutually exclusive, and the union of the events is the entire sample space.

Law of Total Probability

P(A) = ∑(i=1 to n) P(A|B₋i)P(B₋i)

Bayes Rule

The probability of a subset B₋k of a partition, given an event A:

Random Variable

If X is a random variable, then P(X = x) is the probability of seeing a specific value in the sample space.

Discrete Random Variable

A random variable where the range of values that the variable can take on is a countable set

Probability Function

f(x) = P[X = x], x ∈ Rx (the range of values x can take on)

E(x)

Expected Value of a DRV

Var(x)

Variance of a DRV

Bernoulli Random Variable

A random variable that can only take on one of two values.

Trial

each result/observation of an experiment, such as one roll of a number cube.

Binomial Distribution

Used to find the probability of x successes in n trials.

Geometric Distribution

Used to find the probability that the first success happens on the xth trial.

Negative Binomial Distribution

Used to find the probability that the rth success happens on the xth trial.

Hypergeometric Distribution

Used to find the probability of choosing exactly x elements of a given type r out of a population of N elements to fill a subset of n elements.