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Relations and Functions
Relations and Functions
Algebra
Inverse Trigonometric Functions
Matrices
- Concept of Matrices
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- Operations on Matrices> Addition and Subtraction of Matrices
- Operations on Matrices>Scalar Multiplication
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Calculus
Determinants
Vectors and Three-dimensional Geometry
Continuity and Differentiability
- Continuous and Discontinuous Functions
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivatives of Composite Functions
- Derivative of Implicit Functions
- Derivative of Inverse Function
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Overview of Continuity and Differentiability
Linear Programming
Probability
Applications of Derivatives
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Properties of Indefinite Integral
- Methods of Integration> Integration by Substitution
- Methods of Integration>Integration Using Trigonometric Identities
- Methods of Integration> Integration Using Partial Fraction
- Methods of Integration> Integration by Parts
- Integrals of Some Particular Functions
- Definite Integrals
- Fundamental Theorem of Integral Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Overview of Integrals
Sets
Applications of the Integrals
Differential Equations
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
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- Methods of Solving Differential Equations> Variable Separable Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
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- Overview of Differential Equations
Vectors
- Basic Concepts of Vector Algebra
- Direction Ratios, Direction Cosine & Direction Angles
- Types of Vectors in Algebra
- Algebra of Vector Addition
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- Vector Joining Two Points in Algebra
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Three - Dimensional Geometry
Linear Programming
Probability
Introduction
The Multiplication Theorem of Probability is used to find the probability that two or more events occur together. It is closely connected with conditional probability, because the chance of one event may depend on whether another event has already occurred.
This theorem is especially useful in questions involving successive events, such as drawing cards, selecting balls without replacement, or performing actions step by step.
Maharashtra State Board: Class 12
Theorem: Multiplication Theorem
For two events:
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\[P(E \cap F) = P(F) \cdot P(E | F)\]
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\[P(E \cap F) = P(E) \cdot P(F | E)\]
For three events:
- \[P(E \cap F \cap G) = P(E) \cdot P(F | E) \cdot P(G | E \cap F)\]
When to Use This Theorem
Example 1
An urn contains 10 black balls and 5 white balls. Two balls are drawn one after another without replacement. Find the probability that both balls are black.
Step 1: Define events
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Let E = {first ball is black.}
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Let F = {second ball is black.}
Step 2: Find probabilities
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\[P(E) = \frac{10}{15}\]
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After one black ball is removed, 9 black balls remain out of 14 balls. Therefore, \[P(F | E) = \frac{9}{14}\].
Step 3: Apply multiplication theorem
Answer: Probability that both balls are black = \[\frac{3}{7}\].
Real Life Examples
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If a student attends class regularly and then prepares properly, the probability of scoring well in a test can be thought of as a sequence of related events.
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In a cricket match, the chance that a team wins after a strong opening partnership can be viewed through conditional probability.
Key Points: Multiplication Theorem on Probability
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Multiplication theorem is used to find the probability of simultaneous occurrence of events.
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For two events: \[P(E \cap F) = P(E) \cdot P(F | E)\]
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Another equivalent form is \[P(E \cap F) = P(F) \cdot P(E | F)\].
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For three events: \[P(E \cap F \cap G) = P(E) \cdot P(F | E) \cdot P(G | E \cap F)\].
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Most “without replacement” questions are solved using this theorem.
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Always define events before solving a probability problem.
