Probability

1. Overview

Probability is a measure quantifying the likelihood that events will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, where, roughly speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).

These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in such areas of study as mathematics, finance, gambling, science (in particular physics), artificial intelligence/machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.

2. Description

2.1 Definition

2.1.1 Event

$$A$$

2.1.2 Probability

$$P(A)=\frac{prefered}{all}=\frac{favourable}{sample\: space}$$

2.1.3 Trial

Flip and record outcome

2.1.4 Experiment

Multiple trials

2.1.5 Experimental probability

$$P(A)=\frac{successful\:\: trials}{all\:\: trials}$$

2.1.6 Theoretical probability

$$P(A)=\frac{number\: of\: favorable\: outcomes}{all\: trials}$$

2.1.7 Expected value

$$E(A)$$

The outcome we expect to occur when we run an experiment

2.1.8 Probability frequency distribution

• A collection of the probabilities for each possible outcome
• Usually the highest bars will form around the expected value

2.1.9 Complement

$${A}'$$

Everything the event is not

Note 1: ${\left ( {A}' \right )}'=A$

Note 2: $P(A)+P(B)+P(C)=1$

${A}'=B+C$

$P({A}')=1-P(A)$

3. Example

3.1 Experiment and trial

3.2 Expected value

3.3 Complement

4. Reference

https://365datascience.com/

https://en.wikipedia.org/wiki/Probability

https://en.wikipedia.org/wiki/Likelihood_function

https://en.wikipedia.org/wiki/Game_theory