- https://en.wikipedia.org/wiki/Probability#:~:text=Probability%20is%20the%20branch%20of,event%20and%201%20indicates%20certainty.
- Morin, D. (2016). Probability for the Enthusiastic Beginner. Las Vegas, Nevada: CreateSpace Independent Publishing Platform.
- https://en.wikipedia.org/wiki/Permutation

Probability References

- Combinatorics
- Probability Distributions
- Random Variables
- Independence
- Chain Rules

Topics in Probability Theory

This example demonstrates how to calculate the probability of a specific outcome from multiple independent events. Consider a scenario where three friends each flip a fair coin once, and the question is: what is the probability that exactly one of them flips heads? 

To solve this, we first identify the sample space, which consists of all possible outcomes. Since each of the three flips has two outcomes (Heads or Tails), there are $2^3 = 8$ total possibilities: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. 

Next, we identify the favorable outcomes that match the condition of having exactly one head: HTT, THT, and TTH. There are 3 such outcomes. 

The probability is the ratio of favorable outcomes to the total number of outcomes, which is 3/8.

Calculating the Probability of Exactly One Head in Three Coin Flips

The binomial probability formula is used to determine the probability of obtaining exactly *k* successes in a sequence of *n* independent trials. The formula is expressed as $\binom{n}{k} p^k (1-p)^{n-k}$, where *p* represents the probability of success on a single trial, and the binomial coefficient $\binom{n}{k}$ calculates the number of ways to choose *k* successes from *n* trials.

Binomial Probability Formula

A person rolls two standard, fair six-sided dice at the same time. What is the probability that the sum of the numbers shown on the top faces is exactly 5?

Based on your understanding of probability, analyze your friend's claim and conclusion. Is this single experiment of 10 flips sufficient evidence to conclude the coin is not a standard, fair coin? Explain your reasoning.

Evaluating a Claim about a 'Lucky' Coin

A fair spinner is divided into four equal sections, colored Red, Blue, Green, and Yellow. If the spinner is spun three independent times, what is the probability that it lands on 'Red' exactly twice? Explain your reasoning by identifying the total number of possible outcomes and the number of favorable outcomes.

Calculating Probability of Compound Events

Surprisal quantifies the level of surprise experienced by an observer upon witnessing an event $$j$$ to which they had assigned a subjective probability $$P(j)$$. It is defined mathematically as:
$$-\log P(j)$$
Events with lower probabilities yield higher surprisal, reflecting that they convey more information when they occur.

Surprisal

In probability theory, a sample space is the set of all possible values or outcomes that a random observation can take. It serves as the foundational set from which subsets, known as events, are derived.

Sample Space

Probability is the branch of math concerned with the likelihood of an event occuring, or the likelihood that an event is true. 

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Pure:
- Quantity (number theory)
- Space (geometry)
- Structure (Algebra)
- Change (mathematical analysis)
- Combinatorics
- Topology

Applied:
- Probability Theory
- Computational Science
- Mathematical Physics
- Operational Research

Focuses of Math

Statistics is the process of gathering, analyzing, and interpreting information in order to help people make decisions when faced with uncertainty.

Statistics

https://en.wikipedia.org/wiki/Probability#:~:text=Probability%20is%20the%20branch%20of,event%20and%201%20indicates%20certainty.

Probability - Wikipedia

Reference of Foundations of Large Language Models Course

A set that has additional features (e.g. operations, relations, metrics, topology)

Structure in math

Study of financial risk and uncertainty. Combines elements of Business, Data Science, Statistics, and CS to help quantify financial risk. Actuaries are mostly hired into the Financial or Insurance industries to model various economic uncertianties.

Actuarial Mathematics

The branch of "pure" math that studies integers and integer-value functions

Quantity (number theory)

Probability Theory

The list of helpful broad references about Statistics is categorized as follows:
- Statistics books
- Statistical Software 
- Statistics online courses
- Statistics research papers

Statistics References

The marriage of Statistics, Computer Science, and Domin knowledge built Data Science.

Data science is interdisciplinary

A few of the common statistical paradoxes are:"

- Monty Hall's paradox 
- Berkson's paradox
- Simpson's paradox

Common statistical paradoxes

Bayesian statistics is an approach to inferential statistics where a researcher specifies the prior probability that the null hypothesis and alternative hypotheses are true before conducting a study. Once the study is conducted, the probabilities are updated based on the newly collected data. This approach offers a potential alternative to traditional null hypothesis testing by providing a logical framework for quantifying uncertainty using probability distributions.

Bayesian Statistics

Components of Statistical Analysis

Descriptive statistics are a set of quantitative techniques used to summarize, organize, and visually display the data collected from a research sample. Important examples include measures of central tendency (such as the mean, median, and mode), which describe average scores, and measures of dispersion (such as the range, standard deviation, and variance), which indicate how spread apart the scores are.

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