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


Probability is a branch of mathematics that studies the likelihood of events occurring. This concept helps us predict how likely certain events are to occur based on available data or theoretical models. From deciding what to wear based on the weather forecast to assessing risks in various financial sectors, probability plays a vital role in our daily lives.

What is probability?

Probability is a measure between 0 and 1 that tells how likely an event is to occur. A probability of 0 means the event will not occur, while a probability of 1 means the event will definitely occur. Most probabilities lie somewhere in between.

Mathematically, probability can be defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. It can be written as:

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

Basic example

For example, consider the simple event of tossing a coin. What is the probability of getting heads?

  • Number of favourable outcomes (getting heads) = 1
  • Total number of possible outcomes (heads and tails) = 2
    Probability of getting head = 1/2 = 0.5
    Probability of getting head = 1/2 = 0.5

This means that the probability of getting heads when tossing a coin is 50%.

Visualization of probability

Consider a six-sided die. Each side has a number from 1 to 6. What is the probability of getting a 3?

1 2 3 4 5 6

Out of these 6 possible outcomes, getting 3 is just one outcome. So, the probability is:

    Probability of rolling a 3 = 1/6 ≈ 0.1667
    Probability of rolling a 3 = 1/6 ≈ 0.1667

This probability, approximately 16.67%, indicates that the theoretical expectation is that 3 will appear approximately 16 times in 100 rolls.

Types of probability

Probability can be classified into several types:

Theoretical probability

Theoretical probability is based on the logic behind probability. It helps determine how likely a specific event is to occur, based only on the possible outcomes. For example, the probability of drawing an ace from a standard deck of 52 cards can be calculated as follows:

    Probability of drawing an Ace = Number of Aces / Total cards = 4/52 = 1/13 ≈ 0.0769
    Probability of drawing an Ace = Number of Aces / Total cards = 4/52 = 1/13 ≈ 0.0769

Experimental probability

Experimental probability is determined through actual experiments and is based on the number of possible outcomes multiplied by the total number of trials. For example, if you roll a dice 100 times and the number 3 comes up 18 times, the experimental probability is:

    Experimental probability = Number of times event occurs / Total number of trials Probability of rolling 3 = 18/100 = 0.18
    Experimental probability = Number of times event occurs / Total number of trials Probability of rolling 3 = 18/100 = 0.18

Note that experimental probabilities may vary between trials due to the random nature.

Subjective probability

Subjective probability is based on intuition and personal judgment. It is not based on physical evidence or experiments. For example, estimating the probability of rain tomorrow may be based on your knowledge of weather patterns, not on exact calculations.

Laws of probability

Here are some fundamental rules you should know about probability:

Law of complementary events

The law of complementary events states that the sum of the probabilities of an event and its complement is equal to 1. If P(A) is the probability of an event, then the probability of the event not occurring, P(A c ) is:

    P(A c ) = 1 - P(A)
    P(A c ) = 1 - P(A)

For example, the probability of not getting a 6 on the die is:

    P(not 6) = 1 - P(6) = 1 - 1/6 = 5/6
    P(not 6) = 1 - P(6) = 1 - 1/6 = 5/6

Sum rule of probability

The sum rule is used to find the probability of the union of two events. If A and B are two events, then the probability of A or B occurring (or both) is calculated as:

    P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
    P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

For example, consider drawing a card from a deck. Let A be the event that the card is a heart and B be the event that it is an ace. The probability of drawing a heart or an ace is calculated by taking into account that the ace of hearts is counted twice:

    P(Heart ∪ Ace) = P(Heart) + P(Ace) - P(Heart ∩ Ace) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13
    P(Heart ∪ Ace) = P(Heart) + P(Ace) - P(Heart ∩ Ace) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13

Law of multiplicative probability

This rule is applied to find the probability of intersection of two dependent events. If A and B are two events, then:

    P(A ∩ B) = P(A) × P(B|A)
    P(A ∩ B) = P(A) × P(B|A)

where P(B|A) is the probability of B, given that A has already occurred.

Independent and dependent events

It is important to understand whether the events are independent or dependent in order to correctly calculate probabilities.

Independent events

Events are independent if the occurrence of one event has no effect on the occurrence of another. For example, tossing a coin and throwing a dice at the same time are independent events. The probability of both outcomes (rolling heads and 4) is:

    P(Heads and 4) = P(Heads) × P(4) = 1/2 × 1/6 = 1/12
    P(Heads and 4) = P(Heads) × P(4) = 1/2 × 1/6 = 1/12

Dependent events

Events depend on whether the outcome or occurrence of the first event affects the outcome or occurrence of the second event in some way. For example, drawing two cards from a deck without replacement. The probability of drawing a king followed by a queen is:

    P(Queen ∩ King) = P(Queen) × P(King|Queen) = 4/52 × 4/51 = 1/13 × 4/51 = 4/663
    P(Queen ∩ King) = P(Queen) × P(King|Queen) = 4/52 × 4/51 = 1/13 × 4/51 = 4/663

Conclusion

Probability is an important tool in the mathematical toolkit for understanding and dealing with uncertainty. Using principles such as the probability of events, different types of probabilities, and the rules governing overlap and intersection, you can effectively tackle a wide range of problems in various areas of life and work. Combining both theoretical and methodological understanding, probability allows us to assess and make predictions about the world around us with reasonable certainty.

Practice and explore

To build a strong intuition for probability, continue to practice problems, perform your own small experiments, and use real-world data to calculate probabilities. The more you practice, the more naturally probability will become part of your analytical skill set.


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