Probability Theory

Introduction to probability, sets and set notation, random experiments

What is ground truth?

Gathering data to identify the true answer to a question (in psych research) 

What is data used to do?

Model (formal rep of a system) the world

What are deterministic models?

• Don’t take into account other (unanticipated) factors/exceptions 

• Imply certainty + consistency BUT real world data + human subjects are complex

  • Many factors we can’t anticipate for in studies

  • Incomplete info

  • Problem of induction

What are probabilistic/stochastic models?

• Inferential statistics

  • Tell you something about the probability of your data

  • Helps guide your decision about the ground truth 

• Take element of randomness into account 

What is probability?

• Likelihood of event's occurrence (0-1 = impossible vs certain)

  • Number of ways x can happen divided by the number of possible outcomes (including x) → must find out these values 

• Analytic definition 

  • The probability of an event is equal to the ratio of successful outcomes to all possible outcomes 

How does relative frequency relate to probability?

• The probability of an event is the proportion of times you would observe x if you took an infinite number of samples (e.g. 1/6 to get a 6 on a dice) 

• Law of large numbers = given an even + a probability, over n trials, the probability that the relative frequency of x will differ from P(x) approaches 0 as n approaches infinity

What is a set?

Well-defined collection of objects; composed of elements/members 

What is the notation for elements in a set?

What is a subset?

All elements of B must also be in A BUT all elements in A don’t have to exist in B

What are the notations for subsets?

What is the notation for the complement of A?

What terms are used to describe how distinct sets interact with each other?

• Union

• Intersection

• Difference

• Empty set

What is a union?

Element is a member of set A or set B or both 

What is an intersection?

Element is a member of set A and B 

What is the code for calculating the intersection between sets?

What is ‘difference’ in probability sets?

An element is a member of set A BUT NOT set B 

What is the code for calculating difference between sets?

What is a universal set?

All possible elements in a category of interest

What is an empty set? What is the equivalent of this?

No elements (e.g. intersection of mutually exclusive sets → when one occurs, the other can't) 

What are random experiments?

• Can be repeated infinitely under identical conditions 

• Outcome depends on chance + can't be determined in advance 

• Can make inferences about the likelihood of each of its outcomes (sample space) 

What is a simple event?

A single element in a sample space

What is a probability distribution?

Maths function that describes the probability of each event within the sample space 

• Allows you to visualise the likelihood of all possible outcomes