Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Probability and Statistics → Probability ↓


Combinatorial Probability


Combinatorial probability is a fascinating part of probability and statistics. It deals with situations where we need to count the number of ways in which things can happen and then calculate the probability of these events. This concept relies heavily on understanding combinations and permutations.

Understanding combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging and combining objects. It answers questions such as "In how many different ways can these events occur?". It forms the basis of combinatoric probability.

Let us first understand what permutations and combinations are:

Permutation

Permutation is arranging objects in a specific order. For example, if we have three letters A, B and C, we can arrange them in different orders:

        1. ABC
        2. ACB
        3. BAC
        4. BCA
        5. CAB
        6. CBA

In general, for n unique objects, there are n! (n factorial) permutations. The factorial of a non-negative integer ( n ), denoted by ( n ), is the product of all positive integers less than or equal to ( n ).

Combination

Combinations are selections of items where the order does not matter. For example, if you want to select 2 out of 3 letters A, B and C, you have the following combinations (the order is not important):

        1. AB
        2. AC
        3. BC

To calculate the combinations we use the formula:

C(n, r) = n! / (r! * (n-r)!)

Where n is the total number of items, and r is the number of items to choose from.

Calculating probability using combinatorics

Probability is a measure of the likelihood of an event occurring. It is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes.

Mathematically this can be expressed as:

P(Event) = Number of Favorable Outcomes / Total Number of Possible Outcomes

With combinatorial probability, we often use combinations and permutations to determine the number of favorable and possible outcomes.

Example 1: Lottery ticket

Imagine a lottery game where you have to choose 4 numbers, and each can be from 1 to 10. What is the probability of choosing the numbers 1, 2, 3 and 4 in the same order?

First, determine the total number of possible outcomes. Since the order matters, this is a permutation problem:

Total Outcomes = P(10, 4) = 10! / (10-4)! = 10 * 9 * 8 * 7 = 5040

There is only one way to get the results 1, 2, 3, 4 in that order. So:

P (1, 2, 3, 4) = 1 / 5040 ≈ 0.000198

Example 2: Choosing a team

Suppose you have a group of 12 students, and you have to select a team of 4. What is the number of ways to select a team?

This is a combinational problem because the order in which you choose doesn’t matter. Like this:

C(12, 4) = 12! / (4! * (12-4)!) = 495

Therefore, there are 495 different ways to form a team of 4 students.

Visual representation

Sometimes the use of diagrams or illustrations can help clarify problems.

Tree diagram

A tree diagram can illustrate the counting principle. Let's see how a tree diagram shows the number of ways to choose 2 items from a set {A, B, C} regardless of the order.

A B A,B A,C B,C B.A

The same combination is shown more than once in the picture, as there is no specific order in this example.

Application of combinatorial probability

You can see combinatorial probability everywhere from games to real-life scenarios. Here are some everyday examples:

Example 3: Card deck

Consider a standard deck of 52 cards. What is the probability of getting 4 aces?

First, find the number of combinations of getting 4 aces:

C(4, 4) = 4! / (4! * (4-4)!) = 1

Then, count the number of ways to choose 4 cards from the whole deck:

C(52, 4) = 52! / (4! * (52-4)!) = 270725

Therefore, the probability is:

P (4 aces) = 1 / 270725 ≈ 0.0000037

Example 4: Simple lottery

In a simple lottery with numbers from 1 to 100, if you have to pick the winning numbers, what is the probability of guessing correctly?

There is only one favorable outcome when you pick the right numbers:

The total possible outcomes are 100, as there are 100 different numbers.

So:

P (Winning) = 1 / 100 = 0.01 or 1%

Conclusion

Combinatorial probability provides a structured way of looking at the likelihood of events occurring by combining basic counting principles and probability theory. Understanding how permutations and combinations are calculated is essential and opens up different analytical strategies for dealing with uncertainty.

Whether you are considering simple scenarios, such as drawing a card from a deck, or more complex cases, these concepts form the basis for making informed predictions about random events. They are important not only in school mathematics, but also in fields as diverse as engineering, economics, biology, and artificial intelligence.


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