Algebra

• Arithmetic & Geometric Series
• Binomial Theorem
• Completing The Square

Functions

• Composite Function
• Concept And Notation
• Exponentials & Logarithms
• Factor Theorem
• Inverse Function
• Quadratic Functions
  • Quadratic Forms
• Transformations

Trig

• Area of Sectors
• Circular (Trig) Functions
• Oblique Triangles
• Radian Measure
• Right Angle Trig
• Right Angle Trig on the Unit Circle
• Tangent and Slope of Lines
• Trig Identities

Matrices

• Determinants and Inverses
• Matrix Definitions
• Matrix Operations
• System of Equations

Vectors

• 2D and 3D Lines
• Basic Vector Arithmetic
• Intro To Vectors
• More on Lines
• Scalar Product

Stats

• Continuous Variables
• Discrete Variables
• Standard Deviation
• Statistics Representations
• StatsVocabulary
• Using Normal Distributions

Probability

• Binomial Distribution
• Probability Rules
• Representations of Probability

Calculus

• Derivatives by First Principle
• Intro to Derivatives
  • Derivatives Of More Complex Functions
  • Derivatives Part 1
  • Derivatives Part 2
• Intro To Integration
  • Indefinite Integrals
• Kinematics
• Limits
• Local Max and Min
• Optimization
• Tangent and Normals
• Volumes of Revolution

We'll build this up with an example. If I flip the coin once there are two possible outcomes: H and T. Each outcome occurs 1 time.

If I flip a coin twice there are 4 possible outcomes HH, TH, HT and TT. Notice that the outcome of 1 head and 1 tail occurs twice (or two different ways). Putting this into a table:

For three flips of a coin there are 4 separate outcomes 3 Heads (1 occurrence), 2 Heads and 1 Tail (3 occurrences), 1 Head and 2 Tail (3 occurrences) and 3 Tails (1 occurrence).

Now if we make a new table with the number of occurrences of each outcome we get something like:

This pattern should look familiar. Its the 2nd through 4th row of Pascals triangle, which is the pattern that binomial expansions follow! Lets expand the table bigger to try to explain a bit more.

If we look at the last row this represents flipping a coin 5 times. This shows us that there are only 1 way of getting 5 heads, 5 ways of getting 4 heads and 1 tails, 10 ways of getting 3 heads and 2 tails, etc…

This is the very same idea as with binomial expansions!

Example 1

If we want to know the probability of a certain outcome, say 4 heads out of 5 flips, first we need to calculate the probability this occurring:

BUT! This can happen 5 different ways for we must multiply by 5 giving a final probability of: $\frac{5}{32}$.

Using the formula from above:

Example 2

Find the probability of rolling a number less than 5 on a die for 3 out of 4 rolls.

First the probability of rolling less than a 5 is $\frac{4}{6}$ and the probability of rolling 5 or more is $\frac {2}{6}$. So the total probability is:

But, there are 4 different ways for this to occur (SSSF, SSFS, SFSS, FSSS - S is for success and F is for failure) so we have to multiply our probability by 4.

Using the equation from above:

Okay, but the 1st Row of Pascal's Triangle is Missing!

This corresponds to $n=0$ and $r=0$. Which in the language of probability means we did zero trials ($n=0$) to which there is only 1 outcome (the value in the 1st row) and that is zero successes ($r=0$).