Probability

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The field of probability, within mathematics, assesses the chances of an event occurring.

This likelihood is represented by a numerical value ranging between 0 and 1, where higher values indicate a greater probability.

Event

Events are sets of outcomes in experiments with assigned probabilities. A single outcome can belong to many different events,

which are not equally likely since they can include different groups of outcomes.

Likelihood

A measure of how well a statistical model explains the observed data. In probability,

it often refers to the probability of an interval rather than an individual value.

Outcome

A single result from a probability experiment. Each distinct outcome has a probability assigned to it based on certain criteria,

such as the size of its associated interval in a probability distribution.

Preferred Outcomes

Preferred outcomes are the outcomes we want to occur or the outcomes we are interested in. We also call refer to such

outcomes as “Favorable”.

Sample Space

Sample space refers to all possible outcomes that can occur. Its “size” indicates the amount of elements in it.

Probability Formula

The Probability of event X occurring equals the number of preferred outcomes over the number of outcomes in the

sample space.

Trial

Observing an event occur and recording the outcome.

Experiment

A collection of one or multiple trials.

Experimental Probability

The probability we assign an event, based on an experiment we conduct.

Expected Value

The weighted average of all possible values that a random variable can take on.

It is the specific outcome we expect to occur when we run an experiment.

Expected value for categorical variables.

Expected Value (Numerical Variables)

Frequency

Frequency is the number of times a given value or outcome appears in the sample space.

Frequency Distribution Table

The frequency distribution table is a table matching each distinct outcome in the sample space to its associated frequency.

Probability Frequency Distribution

A collection of the probabilities for each possible outcome of an event.

Complement

The complement of an event is everything an event is not.

Characteristics of Complements

• Can never occur simultaneously.

• Add up to the sample space. (A + A’ = Sample space)

• Their probabilities add up to 1. (P(A) + P(A’) = 1)

• The complement of a complement is the original event. ((A’)’ = A)

The probability of an event A is 0.25.

What is the probability of A’ (the compliment of A)?

P(A’) = 0.75

Combinatorics

A branch of mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints.

Factorials

Factorials express the product of all integers from 1 to n and we denote them with the “!” symbol.

Key Values for Factorials

• 0! = 1.

• If n<0, n! does not exist.

Rules for Factorial Multiplication. (For n>0 and n>k)

Natural Numbers

The set of positive integers (1, 2, 3, ...) that are used in counting. Factorials are computed using these numbers.

Permutations

Permutations represent the number of different possible ways we can arrange a number of elements.

Characteristics of Permutations

• Arranging all elements within the sample space.

• No repetition.

• Pn = n × (n − 1) × (n − 2) × ⋯ × 1 = n! (Called “n factorial”)

In how many ways can we arrange 5 people?

120

Variations

Variations represent the number of different possible ways we can pick and arrange a number of elements.

Variations (With Repetition)

Variations (Without Repetition)

Combinations

Combinations represent the number of different possible ways we can pick a number of elements.

Characteristics of Combinations

• Takes into account double-counting.

• All the different permutations of a single combination are different variations.

• Combinations are symmetric, so C(p, n) = C((n−p), n) , since selecting p elements is the same as omitting n-p elements.

Combinations (Separate Sample Space)

Characteristics of Combinations (Separate Sample Space)

• The option we choose for any element does not affect the number of options for the other elements.

• The order in which we pick the individual elements is arbitrary.

• We need to know the size of the sample space for each individual element.

Combinations with Repetition

Combinations represent the number of different possible ways we can pick a number of elements. In special

cases we can have repetition in combinations and for those we use a different formula.

Set

A set is a collection of elements, which hold certain values. Additionally, every event has a set of outcomes that satisfy it.

The Null-Set

The null-set (or empty set), denoted ∅, is an set which contain no values.

Finite Set

A set with a limited number of elements, which is a fundamental concept in combinatorics.

𝑥 ∈ A

Element x is a part of set A.

A ∋ 𝑥

Set A contains element x.

𝑥 ∉ 𝐴

Element x is NOT a part of set A.

∀𝑥:

For all/any x such that…

A ⊆ B

A is a subset of B.

Intersection (A ∩ B)

The intersection of two or more events expresses the set of outcomes that satisfy all the events simultaneously.

Graphically, this is the area where the sets intersect.

Union (A ∪ B)

The union of two or more events expresses the set of outcomes that satisfy at least one of the events.

Graphically, this is the area that includes both sets.