Mathematics

Introduction to Probability 1. Basic Probability Probability Scale Probability is measured on a scale from $0$ to $1$: $0$: Impossible event. $1$: Certain event. $0.5$: Equally likely as not. Fundamental Rules Complementary Events: The probability of an event NOT happening is $P(\text{not } A) = 1 - P(A)$. Relative Frequency: An estimate of probability based on experimental data. $$\text{Relative Frequency} = \frac{\text{Number of successful trials}}{\text{Total number of trials}}$$ Expected Frequency: The number of times an event is expected to occur over a number of trials. $$\text{Expected Frequency} = P(A) \times \text{Total Trials}$$ 2. Sample Space A Sample Space is a list of all possible outcomes of an experiment. ...

Perimeter, Area, and Volume 1. 2D Shapes Area and Perimeter Shape Perimeter Area Rectangle $2(l + w)$ $l \times w$ Triangle - $\frac{1}{2} \times \text{base} \times \text{height}$ Parallelogram - $\text{base} \times \text{height}$ Trapezium - $\frac{1}{2}(a + b)h$ Circles Circumference: $C = 2\pi r$ or $\pi d$. Area: $A = \pi r^2$. Arc Length: $L = \frac{\theta}{360} \times 2\pi r$. Sector Area: $A = \frac{\theta}{360} \times \pi r^2$. For major sectors, $\theta$ is the reflex angle. ...

Equations, Factors, and Formulae 1. Equations Linear Equations in One Unknown Solving for a variable by isolating it using inverse operations: Addition $\leftrightarrow$ Subtraction Multiplication $\leftrightarrow$ Division Squaring $\leftrightarrow$ Square Root Simultaneous Linear Equations Finding a common solution $(x, y)$ for two linear equations. Substitution Method: Express one variable in terms of the other from one equation, then substitute into the second. Elimination Method: Multiply equations to make coefficients of one variable equal (or opposite), then add or subtract the equations to eliminate that variable. ...

Fractions, Percentages, and Standard Form 1. Fractions Definitions Proper Fraction: Numerator is smaller than denominator (e.g., $1/2, 3/4$). Improper Fraction: Numerator is greater than or equal to denominator (e.g., $5/4, 7/3$). Mixed Number: Combination of a whole number and a proper fraction (e.g., $1 \frac{1}{4}$). Conversions Mixed Number $\rightarrow$ Improper Fraction: $\text{Whole} \times \text{Denominator} + \text{Numerator} / \text{Denominator}$. Improper Fraction $\rightarrow$ Mixed Number: Divide numerator by denominator; quotient is whole number, remainder is numerator. Basic Operations Addition/Subtraction: Find a common denominator, then add/subtract numerators. Example: $\frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12}$ Example: $\frac{3}{4} - \frac{1}{6} = \frac{9}{12} - \frac{2}{12} = \frac{7}{12}$ Multiplication: Multiply numerators together and denominators together. Example: $\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$ Division: Multiply by the reciprocal of the divisor (Keep-Change-Flip). Example: $\frac{2}{3} \div \frac{5}{7} = \frac{2}{3} \times \frac{7}{5} = \frac{14}{15}$ 2. Percentages Percentage of a Quantity: $\frac{\text{percentage}}{100} \times \text{quantity}$. Percentage Increase/Decrease: $\text{Change} = \frac{\text{percentage}}{100} \times \text{original}$. $\text{New Value} = \text{original} \pm \text{change}$. Simple Interest: $I = P \times R \times T$ (Principal $\times$ Rate $\times$ Time). Compound Interest: $A = P(1 + \frac{r}{100})^n$ (Total amount after $n$ years). Reverse Percentages: Finding the original value after a percentage change. $\text{Original} = \frac{\text{New Value}}{1 \pm \frac{\text{percentage}}{100}}$. Repeated Percentage Change: Applying a percentage change multiple times sequentially. Exponential Growth and Decay: Growth: $y = a(1 + r)^x$ Decay: $y = a(1 - r)^x$ Where $a$ is initial value, $r$ is rate, and $x$ is time. ...

Collection, Organising, and Displaying Data 1. Collection and Classifying Data Data can be categorized into two main types: Qualitative (Categorical): Non-numerical data. Nominal: Categories with no inherent order (e.g., eye color, country). Ordinal: Categories with a logical order (e.g., satisfaction rating: low, medium, high). Quantitative (Numerical): Numerical data. Discrete: Values that can only take specific values, often counts (e.g., number of students in a class). Continuous: Values that can take any value within a range, often measurements (e.g., height, weight, time). 2. Organising Data Before displaying data, it must be organized to make analysis easier. ...