Grade 12
  1. Algebra  1.1. Matrices  1.1.1. Definition and Types of Matrices  1.1.2. Operations on Matrices  1.1.3. Transpose of a matrix  1.1.4. Determinants and their properties  1.1.5. Inverse of a Matrix  1.1.6. Solving Linear Equations Using Matrices  1.1.7. Applications of Matrices  1.2. Complex Numbers  1.2.1. Definition and representation  1.2.2. Algebra of complex numbers  1.2.3. Modulus and argument  1.2.4. Polar and Exponential Forms  1.2.5. De Moivre's Theorem  1.2.6. Roots of complex numbers  1.2.7. Applications in Geometry  1.3. Vectors and 3D Geometry  1.3.1. Vector Algebra  1.3.2. Scalar and Vector Products  1.3.3. Equation of a Line in Space  1.3.4. Equation of a Plane  1.3.5. Distance between lines and planes  1.3.6. Angle between Lines and Planes
  2. Calculus  2.1. Limits and Continuity  2.1.1. Concept of Limits  2.1.2. Continuity of Functions  2.1.3. Types of Discontinuities  2.1.4. Left-hand and Right-hand Limits  2.2. Differentiation  2.2.1. Derivative as a Rate of Change  2.2.2. Techniques of Differentiation  2.2.3. Higher Order Derivatives  2.2.4. Applications of Derivatives  2.2.5. Tangents and Normals  2.2.6. Increasing and Decreasing Functions  2.3. Integration  2.3.1. Indefinite Integrals  2.3.2. Definite Integrals  2.3.3. Integration Techniques  2.3.4. Area under a curve  2.3.5. Applications of Integrals in Geometry  2.4. Differential Equations  2.4.1. Formation of Differential Equations  2.4.2. Order and Degree of Differential Equations  2.4.3. Solutions of Differential Equations  2.4.4. Applications of Differential Equations  2.4.5. Solving First-Order Linear Differential Equations
  3. Probability and Statistics  3.1. Probability  3.1.1. Conditional Probability  3.1.2. Bayes' Theorem  3.1.3. Random Variables  3.1.4. Probability Distributions  3.1.5. Binomial and Normal Distributions  3.2. Statistics  3.2.1. Mean and Standard Deviation  3.2.2. Variance and Covariance  3.2.3. Correlation and Regression  3.2.4. Hypothesis Testing
  4. Linear Programming  4.1. Graphical Method  4.1.1. Feasible and Infeasible Regions  4.1.2. Optimal solutions  4.1.3. Sensitivity Analysis in Graphical Method  4.2. Simplex Method  4.2.1. Formulation of Problems  4.2.2. Solving linear programming problems  4.2.3. Dual Simplex Method
  5. Relations and Functions  5.1. Types of Relations  5.1.1. Reflexive Relations  5.1.2. Symmetric Relations  5.1.3. Transitive Relations  5.1.4. Equivalence Relations  5.2. Types of Functions  5.2.1. Onto Functions  5.2.2. One-to-One Functions  5.2.3. Inverse Functions  5.2.4. Composite Functions  5.3. Inverse Trigonometric Functions  5.3.1. Principal Values in Inverse Trigonometric Functions  5.3.2. Properties and Graphs of Inverse Trigonometric Functions  5.3.3. Applications in Calculus
  6. Advanced Trigonometry  6.1. Trigonometric Equations  6.1.1. General solutions of equations  6.1.2. Transformation formulae  6.2. Hyperbolic Functions  6.2.1. Definitions and properties  6.2.2. Relationships between hyperbolic and trigonometric functions
  7. Mathematical Reasoning  7.1. Logical Reasoning  7.1.1. Statements and Logical Connectives  7.1.2. Tautologies and Contradictions  7.2. Proofs  7.2.1. Direct proof  7.2.2. Indirect Proof  7.2.3. Proof by contradiction  7.2.4. Proof by Mathematical Induction
  8. Applications of Mathematics  8.1. Applications of Calculus  8.1.1. Maxima and Minima Problems  8.1.2. Economics and Biology Applications  8.1.3. Rate of Change in Physics  8.2. Applications of Probability  8.2.1. Risk Analysis  8.2.2. Decision Making  8.2.3. Game Theory

Grade 12 ↓


Probability and Statistics


Probability and statistics are two interrelated fields of mathematics that deal with analyzing the probability of events and interpreting data. In this context, we will explore the key concepts, definitions, theorems, and examples using these two branches.

Introduction to probability

Probability is a measure of the likelihood of an event occurring. It is measured as a number between 0 and 1, where 0 represents impossibility and 1 represents certainty. The higher the probability of an event, the more likely it is to occur.

Example: If you flip a fair coin, the probability of getting heads is 0.5. This is because the coin has two sides (heads and tails), and both outcomes are equally likely.

Basics of probability

To understand probability, we start with some basic terms:

  • Experiment: A situation involving chance or probability that leads to results called outcomes. For example, throwing dice is an experiment.
  • Outcome: The possible result of a probability experiment. For a die, the possible outcomes are 1, 2, 3, 4, 5, and 6.
  • Event: A specific outcome or a set of outcomes. For example, getting an even number (2, 4, or 6) is an event.
  • Sample space: The set of all possible outcomes. If you throw a dice, the sample space is {1, 2, 3, 4, 5, 6}.

The probability of an event occurring is calculated by dividing the number of favorable outcomes by the total number of possible outcomes in the sample space.

Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes)

Types of probability

  • Theoretical probability: This is determined based on logic. If we have a fair dice, each side has a 1 in 6 chance of falling face up.
  • Experimental probability: This is determined based on actual experiments. For example, if you flip a coin 100 times and it comes up heads 53 times, the experimental probability of getting heads is 53/100.
  • Subjective probability: It is based on personal judgment. For example, the probability of rain tomorrow can be subjective based on the current weather conditions and past experiences.

Visual example: coin toss

Results: Top

This simple visualization shows the outcome of a coin toss.

Introduction to statistics

Statistics is the branch of mathematics that deals with collecting, analyzing, interpreting, presenting, and organizing data. It allows us to understand data and make decisions based on it.

Data storage and types

Data can be collected in a variety of ways, and it is usually classified into two types:

  • Qualitative data: This refers to descriptive data that cannot be measured but can be classified. Examples include colors, names, or labels.
  • Quantitative data: This refers to numerical data that can be measured. Examples include age, height, or salary.

Organizing the data

Organizing data involves summarizing and presenting it so that useful information can be extracted. This can be done using tables, charts or graphs.

Example: A table showing the number of students in three different classes:

Class | Number of Students --------------------------- A | 30 B | 25 C | 35

Measures of central tendency

Measures of central tendency are statistical measurements that show the center or typical value of a dataset. The most common are the mean, median, and mode.

  • Mean: The average of a dataset. It is calculated by adding up all the data points and dividing by the number of data points. 
    Mean = (Sum of all data points) / (Number of data points)
  • Median: The middle value of a dataset when it is sorted. If the number of data points is odd, the median is the middle one. If the number is even, it is the average of the two middle points.
  • Mode: The data point that appears most often in the dataset.

Example: Consider the data set: 2, 3, 3, 6, 7.

  • Mean: (2 + 3 + 3 + 6 + 7) / 5 = 4.2
  • Median: 3 (middle value)
  • Mode: 3 (most frequently occurring value)

Measures of dispersion

These measures provide an estimate of how spread out the data points are in the dataset.

  • Range: The difference between the highest and lowest data points in a dataset.
  • Variance: The average of the squared differences from the mean.
  • Standard deviation: The square root of the variance, which gives a measure of the average distance from the mean.

Example: Consider the data set: 5, 6, 8, 9, 10.

  • Range: 10 – 5 = 5
  • Variance: Square each deviation: (5 - 7.6) 2 , (6 - 7.6) 2 , etc., find their average.
  • Standard deviation: The square root of the variance.

Visual example: bar graph

a class Class B Class C

This bar graph shows the number of students in three classes.

The concept of probability in statistics

In statistics, probability is used to determine how likely an event is to occur given a dataset. It involves applying mathematical rules and formulas to analyze data trends and draw predictions or conclusions.

Conditional probability

Conditional probability refers to the probability of occurrence of an event based on the fact that another event has already occurred.

P(A|B) = P(A ∩ B) / P(B)

where P(A|B) is the conditional probability of event A occurring given that B has occurred, P(A ∩ B) is the probability of both events occurring, and P(B) is the probability of event B.

Example: In a deck of cards, what is the probability that an ace is drawn, given that the card drawn is a spade?

There is an ace of spades in a deck of 52 cards. There are 13 spades in total.

  • P(Ace|Spade) = 1/13

Bayes' theorem

Bayes' theorem provides a way of updating the probability of a hypothesis as more evidence becomes available.

P(A|B) = [P(B|A) * P(A)] / P(B)

This allows statisticians to revise their predictions based on new evidence.

Example: If 1% of a population has a disease, and the test for the disease is 99% accurate, what is the probability that a person has the disease if he or she tests positive?

Plug the values into the formula to find the conditional probability.

Random variables and probability distributions

Random variables are used to measure outcomes. They can be discrete or continuous.

Discrete random variable

These are variables that can take a countable number of values. For example, the number of heads in a coin toss or the roll of a dice.

Continuous random variable

These can take on an infinite number of values within a given range. Examples include time, altitude, and temperature.

Probability distributions

Probability distribution describes how the probabilities are distributed over the values of a random variable.

  • Probability mass function (PMF): applies to discrete random variables and gives the probability that a random variable is exactly equal to some value.
  • Probability density function (PDF): applies to continuous random variables and provides probabilities for a range of values.

Example: Consider rolling a dice, we can chart the PMF.

Roll Possibility
1 1/6
2 1/6
3 1/6
4 1/6
5 1/6
6 1/6

Conclusion

Understanding probability and statistics is crucial for making informed decisions based on data. By applying basic probability concepts, various statistical measures, and theorems such as Bayes' theorem, one can interpret data with greater accuracy. This combination provides powerful tools for defining uncertainty, predicting outcomes, and drawing conclusions from collected data. Thus, these concepts find applications in various fields ranging from science and engineering to economics and social sciences.


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