Probability Formulas and Combinatorics Reference

Posted by on Aug 5, 2026 in Mathematics | 0 comments

Probability Formulas and Counting Techniques

1. Basic Probability Rules (7.2)

  • Probability of an Event: P(A) = (Number of favorable outcomes) / (Total number of outcomes) = n(A) / n(E)

  • Probability Range: 0 ≤ P(A) ≤ 1

    • P(A) = 0 means the event is impossible.

    • P(A) = 1 means the event is certain to occur.

  • Complementary Events (Not A): P(A’) = 1 – P(A) OR P(A) + P(A’) = 1

  • General Addition Formula (For any two events A and B): P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

  • Mutually Exclusive Events: If A and B cannot

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Market Research Methods and Business Strategies

Posted by on Jun 19, 2026 in Mathematics | 0 comments

1. Role of Market Research and Methods Used

Market research is the process of collecting, recording, and analyzing data about customers, competitors, and the market for a product. It helps businesses find out what consumers like or dislike, identify market size, predict future demand changes, and find their unique selling point (USP).

There are two main types of data collected:

  • Quantitative research: Numerical data that can be put into tables, charts, or graphs.
  • Qualitative research: Information about
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Discrete Mathematics: Functions, Groups, and Graph Theory

Posted by on Jun 19, 2026 in Mathematics | 0 comments

Functions and Mapping Properties

Analysis of the Sine Function

a) Let f:R→Rf:\mathbb{R}\rightarrow\mathbb{R}f:R→R be defined by f(x)=sin⁡xf(x)=\sin xf(x)=sinx.

(i) Image Set and Surjectivity

Given f(x)=sin⁡xf(x)=\sin xf(x)=sinx. For every real number xxx, −1≤sin⁡x≤1-1\leq \sin x\leq 1−1≤sinx≤1.

Hence the image set is f(R)={y:y=sin⁡x,  x∈R}f(\mathbb{R})=\{y:y=\sin x,\;x\in\mathbb{R}\}f(R)={y:y=sinx,x∈R} ={y:−1≤y≤1}=\{y:-1\leq y\leq1\}={y:−1≤y≤1}. f(R)=[−1,

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Statistical Analysis and Machine Learning Fundamentals

Posted by on Jun 17, 2026 in Mathematics | 0 comments

Measures of Central Tendency and Dispersion

a) Measures of Central Tendency

These metrics identify the central point of a data distribution.

  • Mean (Average): The sum of all values divided by the count.
    • Example: A retail store counts daily customers over 5 days: [10, 15, 20, 25, 30]. The Mean is (10+15+20+25+30)/5 = 20.
  • Median (Middle Value): The middle number in a sorted list. It is highly resistant to outliers.
    • Example: If software engineer salaries are [$60k, $65k, $70k, $80k, $250k], the Median is $
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Trigonometry, Polar Coordinates, and Vector Analysis

Posted by on Jun 12, 2026 in Mathematics | 0 comments

Trigonometric Functions and Properties

Even Functions: cos(-t) = cos(t), sec(-t) = sec(t)

Odd Functions: sin(-t) = -sin(t), tan(-t) = -tan(t), csc(-t) = -csc(t), cot(-t) = -cot(t)

Example: A point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.

A) P(-15/17, 8/17):

  • sin(t) = 8/17
  • cos(t) = -15/17
  • tan(t) = -8/15
  • csc(t) = 17/8
  • sec(t) = -17/15
  • cot(t) = -15/8

Graphs of Trigonometric Functions

Amplitude, Period, and Phase Shift:

For the equation

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Essential Algebra Formulas and Graphing Techniques

Posted by on May 31, 2026 in Mathematics | 0 comments

Algebraic Formulas and Equations

Quadratic and Polynomial Functions

  • AOS: x = -b/2a
  • Quadratic Formula: x = (-b ± √b² – 4ac) / 2a
  • Discriminant (b² – 4ac):
    • > 0: Two solutions
    • < 0: No real solutions
    • = 0: One solution
  • Completing the Square: Take 1/2 of b, square it, and add to both sides. Rewrite as (x ± 1/2b)².

Sequences and Financial Math

  • Arithmetic Sequence: aₙ = a₁ + d(n – 1)
  • Geometric Sequence: aₙ = a₁ * rⁿ⁻¹
  • Compound Interest (Annually): P(1 + r)ᵗ
  • Compound Interest (Quarterly): P(
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