Redshift Academy

Wolfram Alpha:

Search by keyword:

Astronomy

-

Astronomical Distance Units .

-

Celestial Coordinates .

-

Celestial Navigation .

-

Location of North and South Celestial Poles .

Chemistry

-

Avogadro's Number .

-

Balancing Chemical Equations

-

-

The Periodic Table .

Classical Mechanics

-

Blackbody Radiation .

Classical Physics

-

Archimedes Principle

-

Bernoulli Principle

-

Center of Mass Frame

-

Comparison Between Gravitation and Electrostatics

-

Compton Effect .

-

Coriolis Effect

-

Cyclotron Resonance

-

-

-

Double Slit Experiment

-

Elastic and Inelastic Collisions .

-

Electric Fields

-

-

-

-

Gauss's Law of Universal Gravity .

-

Gravity - Force and Acceleration

-

-

Ideal and Non-Ideal Gas Laws (van der Waal)

-

-

-

-

-

-

Kinetic Theory of Gases .

-

Kirchoff's Laws

-

Maxwell's Equations .

-

Moments and Torque

-

-

One Dimensional Wave Equation .

-

Pascal's Principle

-

Phase and Group Velocity

-

Poiseuille's Law

-

Refractive Index

-

Rotational Dynamics

-

Simple Harmonic Motion

-

Specific Heat, Latent Heat and Calorimetry

-

-

The Laws of Thermodynamics

-

The Zeeman Effect .

-

Young's Modulus

Climate Change

-

Keeling Curve .

Cosmology

-

-

Cosmic Background Radiation and Decoupling .

-

-

-

Friedmann-Robertson-Walker Equations .

-

-

Inflation Theory

-

Introduction to Black Holes .

-

Olbers' Paradox .

-

-

Stephen Hawking's Last Paper .

-

Stephen Hawking's PhD Thesis .

-

The Big Bang Model

-

Vacuum Energy .

Finance and Accounting

-

-

-

Brownian Model of Financial Markets .

-

Capital Structure

-

Dividend Discount Formula

-

Lecture Notes on International Financial Management

-

-

Periodically and Continuously Compounded Interest

-

Repurchase versus Dividend Analysis

Game Theory

-

General Relativity

-

Basis One-forms .

-

Catalog of Spacetimes .

-

Curvature and Parallel Transport

-

Einstein's Field Equations

-

-

Gravitational Waves

-

Hyperbolic Motion and Rindler Coordinates .

-

Quantum Gravity

-

Ricci Decomposition .

-

-

Stress-Energy-Momentum Tensor .

-

-

The Area Metric

-

The Dirac Equation in Curved Spacetime .

-

The Equivalence Principal

-

The Essential Mathematics of General Relativity

-

The Induced Metric

-

The Light Cone .

-

The Metric Tensor .

-

The Principle of Least Action in Relativity .

-

Vierbein (Frame) Fields

Group Theory

-

Basic Group Theory .

-

Basic Representation Theory .

-

Building Groups From Other Groups .

-

Sets, Groups, Modules, Rings and Vector Spaces

-

Symmetric Groups .

-

The Integers Modulo n Under + and x .

Lagrangian and Hamiltonian Mechanics

-

Classical Field Theory .

-

Euler-Lagrange Equation .

-

Ex: Newtonian, Lagrangian and Hamiltonian Mechanics .

-

Hamiltonian Formulation .

-

Liouville's Theorem

-

Symmetry and Conservation Laws - Noether's Theorem .

Macroeconomics

-

Lecture Notes on International Economics

-

Lecture Notes on Macroeconomics

-

Macroeconomic Policy

Mathematics

-

Amplitude, Period and Phase

-

Arithmetic and Geometric Sequences and Series .

-

-

Augmented Matrices and Cramer's Rule

-

Binomial Theorem (Pascal's Triangle)

-

Completing the Square

-

Complex Numbers

-

Composite Functions

-

Conformal Transformations .

-

Conjugate Pair Theorem

-

Contravariant and Covariant Components of a Vector

-

Derivatives of Inverse Functions

-

Double Angle Formulas

-

Eigenvectors and Eigenvalues

-

Euler Formula for Polyhedrons

-

Factoring of a³ +/- b³

-

Fourier Series and Transforms .

-

-

Gauss's Divergence Theorem

-

Grassmann and Clifford Algebras .

-

Heron's Formula

-

Index Notation (Tensors and Matrices) .

-

-

Integration By Parts

-

Introduction to Conformal Field Theory .

-

Inverse of a Function

-

Law of Sines and Cosines

-

Line Integrals, ∮

-

Logarithms and Logarithmic Equations

-

Matrices and Determinants

-

Matrix Exponential

-

Mean Value and Rolle's Theorem

-

Modulus Equations

-

Orthogonal Curvilinear Coordinates .

-

Parabolas, Ellipses and Hyperbolas

-

Piecewise Functions

-

Polar Coordinates

-

Polynomial Division

-

Quaternions 1 .

-

Quaternions 2 .

-

Regular Polygons

-

-

Similar Matrices and Diagonalization .

-

Spherical Trigonometry

-

Stirling's Approximation

-

Sum and Differences of Squares and Cubes

-

-

Tangent and Normal Line

-

Taylor and Maclaurin Series .

-

The Essential Mathematics of Lie Groups

-

The Limit Definition of the Exponential Function

-

Tic-Tac-Toe Factoring

-

Trapezoidal Rule

-

-

Volume Integrals

Microeconomics

-

Marginal Revenue and Cost

Nuclear Physics

-

-

Radioactive Decay

Particle Physics

-

Feynman Diagrams and Loops

-

Field Dimensions

-

Helicity, Chirality and Weyl Spinors .

-

Klein-Gordon and Dirac Equations .

-

Regularization and Renormalization

-

Scattering - Mandelstam Variables

-

Spin 1 Eigenvectors .

Probability and Statistics

-

Box and Whisker Plots

-

Buffon's Needle .

-

Categorical Data - Crosstabs

-

Chebyshev's Theorem

-

Chi Squared Goodness of Fit

-

Conditional Probability

-

Confidence Intervals

-

-

-

Factor Analysis

-

Hypothesis Testing

-

Linear Regression

-

Monte Carlo Methods

-

Non Parametric Tests

-

-

Pearson Correlation

-

Permutations and Combinations

-

Pooled Variance and Standard Error

-

Probability Distributions

-

Probability Rules

-

Sample Size Determination

-

Sampling Distributions

-

Set Theory - Venn Diagrams

-

Stacked and Unstacked Data

-

Stem Plots, Histograms and Ogives

-

Survey Data - Likert Item and Scale

-

-

Programming and Computer Science

-

-

How this site works ...

-

More Programming Topics

-

MVC Architecture

-

Open Systems Interconnection (OSI) Standard - TCP/IP Protocol

-

Public Key Encryption

Quantum Computing

-

Density Operators and Mixed States .

-

Entangled States .

-

Quantum Field Theory

-

Creation and Annihilation Operators

-

Field Operators for Bosons and Fermions

-

Lagrangians in Quantum Field Theory

-

Path Integral Formulation

-

Relativistic Quantum Field Theory

Quantum Mechanics

-

-

Clebsch-Gordan Coefficients .

-

-

-

Electron Orbital Angular Momentum and Spin

-

Heisenberg Uncertainty Principle

-

Ladder Operators .

-

Multi Electron Wavefunctions .

-

Pauli Spin Matrices

-

Photoelectric Effect .

-

Position and Momentum States .

-

Probability Current

-

Schrodinger Equation for Hydrogen Atom .

-

Schrodinger Wave Equation

-

Spin 1/2 Eigenvectors

-

The Differential Operator

-

The Essential Mathematics of Quantum Mechanics

-

The Quantum Harmonic Oscillator .

-

The Schrodinger, Heisenberg and Dirac Pictures

-

The WKB Approximation

-

Time Dependent Perturbation Theory

-

Time Evolution and Symmetry Operations

-

Time Independent Perturbation Theory

-

Semiconductor Reliability

-

The Weibull Distribution

Solid State Electronics

-

Band Theory of Solids .

-

Fermi-Dirac Statistics .

-

Intrinsic and Extrinsic Semiconductors .

-

-

The P-N Junction

Special Relativity

-

-

Electromagnetic (Faraday) Tensor .

-

Energy and Momentum in Special Relativity, E = mc² .

-

Invariance of the Velocity of Light .

-

Lorentz Invariance .

-

Lorentz Transform .

-

Lorentz Transformation of the EM Field .

-

Newton versus Einstein

-

Spinors - Part 1 .

-

Spinors - Part 2 .

-

The Continuity Equation .

-

The Lorentz Group .

Statistical Mechanics

-

Entropy and the Partition Function

-

The Harmonic Oscillator

-

String Theory

-

Bosonic Strings

-

Extra Dimensions

-

Introduction to String Theory

-

Kaluza-Klein Compactification of Closed Strings

-

Strings in Curved Spacetime

-

Toroidal Compactification

Superconductivity

-

Bardeen–Cooper–Schrieffer Theory

-

-

-

Introduction to Superconductivity .

-

Superconductivity (Lectures 1 - 10)

-

Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

-

Chiral Superfields

-

Generators of a Supergroup

-

Grassmann Numbers

-

Introduction to Supersymmetry

-

The Gauge Hierarchy Problem

The Standard Model

-

Electroweak Unification (Glashow-Weinberg-Salam)

-

Gauge Theories (Yang-Mills)

-

Gravitational Force and the Planck Scale

-

Introduction to the Standard Model

-

Isospin, Hypercharge, Weak Isospin and Weak Hypercharge

-

Quantum Flavordynamics and Quantum Chromodynamics

-

Special Unitary Groups and the Standard Model - Part 1 .

-

Special Unitary Groups and the Standard Model - Part 2

-

Special Unitary Groups and the Standard Model - Part 3 .

-

Standard Model Lagrangian

-

The Higgs Mechanism

-

The Nature of the Weak Interaction

Topology

-

Units, Constants and Useful Formulas

-

Last modified: January 26, 2018


Probability Rules
-----------------

Mutually Exclusive Events
-------------------------

A and B cannot happen at the same time.  Therefore:

P(A and B) ≡ P(A ∩ B) = 0

Example:  

A simple coin toss. 

Independent Events
------------------

A and B are not related to each other.  Therefore:

P(A ∩ B) = P(A).P(B)

Clearly A and B cannot be both mutually exclusive and independent.

Example:  

Probability of rolling a 6 and tossing a head is (1/6)(1/2) = 1/2

Dependent Events
----------------

When A and B, are dependent, the probability of both occurring is:
  	
P(A ∩ B)  =  P(A)·P(B|A) 

Example:  

A card is chosen at random from a deck of cards. Without replacing 
it, a second card is chosen. What is the probability that the first 
card chosen is a queen and the second card chosen is a jack? 

P(Q) = 4/52

P(J|Q) = 4/51

P(Q ∩ J) = (4/52)(4/51) = 4/663

Dependent events are discussed in more detail in the note on 
conditional probability.

Addition Rules
--------------

If A and B ARE mutually exclusive:

P(A or B) ≡ P(A ∪ B) = P(A) + P(B)

Example:

A single die is rolled. What is the probability of rolling a 
2 or a 5?

P(2 ∪ 5) = P(2) + P(5)

         = 1/6 + 1/6

         = 1/3

If A and B are NOT mutually exclusive:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

         = P(A) + P(B) - P(A).P(B) 

         = P(A) + P(B)(1 - P(A)) 

Example:  

What is the probability pulling a club or a king from a deck 
of cards?

P(C ∪ K) = P(C) + P(K) - P(C ∩ K)

         = 4/52 + 13/52 - 1/52 or 4/52 + (13/52)(1 - 4/52 )

         = 4/13