#### Topics

##### Mathematical Logic

- Statements - Introduction in Logic
- Sentences and Statement in Logic
- Truth Value of Statement in Logic
- Open Sentences in Logic
- Compound Statement in Logic
- Quantifier and Quantified Statements in Logic
- Logical Connectives
- Truth Tables of Compound Statements
- Examples Related to Real Life and Mathematics
- Statement Patterns and Logical Equivalence
- Algebra of Statements
- Difference Between Converse, Contrapositive, Contradiction
- Application of Logic to Switching Circuits, Switching Table.

##### Mathematical Logic

- Truth Value of Statement in Logic
- Logical Connective, Simple and Compound Statements
- Truth Tables of Compound Statements
- Statement Patterns and Logical Equivalence
- Tautology, Contradiction, and Contingency
- Quantifier and Quantified Statements in Logic
- Duality
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits, Switching Table.

##### Matrics

##### Trigonometric Functions

##### Pair of Straight Lines

##### Vectors

- Representation of Vector
- Vectors and Their Types
- Algebra of Vectors
- Coplanar Vectors
- Vector in Two Dimensions (2-D)
- Three Dimensional (3-D) Coordinate System
- Components of Vector
- Position Vector of a Point P(X, Y, Z) in Space
- Component Form of a Position Vector
- Vector Joining Two Points
- Section formula
- Dot/Scalar Product of Vectors
- Cross/Vector Product of Vectors
- Scalar Triple Product of Vectors
- Vector Triple Product
- Addition of Vectors

##### Line and Plane

##### Linear Programming

##### Matrices

- Elementary Operation (Transformation) of a Matrix
- Inverse by Elementary Transformation
- Elementary Transformation of a Matrix Revision of Cofactor and Minor
- Inverse of a Matrix Existance
- Adjoint Method
- Addition of Matrices
- Solving System of Linear Equations in Two Or Three Variables Using Reduction of a Matrix Or Reduction Method
- Solution of System of Linear Equations by – Inversion Method

##### Differentiation

##### Applications of Derivatives

##### Indefinite Integration

##### Definite Integration

##### Application of Definite Integration

##### Differential Equations

##### Probability Distributions

##### Binomial Distribution

##### Trigonometric Functions

- Trigonometric equations
- General Solution of Trigonometric Equation of the Type
- Solution of a Triangle
- Hero’s Formula in Trigonometric Functions
- Napier Analogues in Trigonometric Functions
- Basic Concepts of Trigonometric Functions
- Inverse Trigonometric Functions - Principal Value Branch
- Graphs of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions

##### Pair of Straight Lines

- Pair of Lines Passing Through Origin - Combined Equation
- Pair of Lines Passing Through Origin - Homogenous Equation
- Theorem - the Joint Equation of a Pair of Lines Passing Through Origin and Its Converse
- Acute Angle Between the Lines
- Condition for Parallel Lines
- Condition for Perpendicular Lines
- Pair of Lines Not Passing Through Origin-combined Equation of Any Two Lines
- Point of Intersection of Two Lines

##### Circle

- Tangent of a Circle - Equation of a Tangent at a Point to Standard Circle
- Tangent of a Circle - Equation of a Tangent at a Point to General Circle
- Condition of tangency
- Tangents to a Circle from a Point Outside the Circle
- Director circle
- Length of Tangent Segments to Circle
- Normal to a Circle - Equation of Normal at a Point

##### Conics

##### Vectors

- Vectors Revision
- Collinearity and Coplanarity of Vectors
- Linear Combination of Vectors
- Condition of collinearity of two vectors
- Conditions of Coplanarity of Three Vectors
- Section formula
- Midpoint Formula for Vector
- Centroid Formula for Vector
- Basic Concepts of Vector Algebra
- Scalar Triple Product of Vectors
- Geometrical Interpretation of Scalar Triple Product
- Application of Vectors to Geometry
- Medians of a Triangle Are Concurrent
- Altitudes of a Triangle Are Concurrent
- Angle Bisectors of a Triangle Are Concurrent
- Diagonals of a Parallelogram Bisect Each Other and Converse
- Median of Trapezium is Parallel to the Parallel Sides and Its Length is Half the Sum of Parallel Sides
- Angle Subtended on a Semicircle is Right Angle

##### Three Dimensional Geometry

##### Line

##### Plane

- Equation of Plane in Normal Form
- Equation of Plane Passing Through the Given Point and Perpendicular to Given Vector
- Equation of Plane Passing Through the Given Point and Parallel to Two Given Vectors
- Equation of a Plane Passing Through Three Non Collinear Points
- Equation of Plane Passing Through the Intersection of Two Given Planes
- Vector and Cartesian Equation of a Plane
- Angle Between Two Planes
- Angle Between Line and a Plane
- Coplanarity of Two Lines
- Distance of a Point from a Plane

##### Linear Programming Problems

##### Continuity

- Introduction of Continuity
- Continuity of a Function at a Point
- Defination of Continuity of a Function at a Point
- Discontinuity of a Function
- Types of Discontinuity
- Concept of Continuity
- Algebra of Continuous Functions
- Continuity in Interval - Definition
- Exponential and Logarithmic Functions
- Continuity of Some Standard Functions - Polynomial Function
- Continuity of Some Standard Functions - Rational Function
- Continuity of Some Standard Functions - Trigonometric Function
- Continuity - Problems

##### Differentiation

- Revision of Derivative
- Relationship Between Continuity and Differentiability
- Every Differentiable Function is Continuous but Converse is Not True
- Derivatives of Composite Functions - Chain Rule
- Derivative of Inverse Function
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Implicit Functions
- Exponential and Logarithmic Functions
- Derivatives of Functions in Parametric Forms
- Derivative of Functions in Product of Function Form
- Derivative of Functions in Quotient of Functions Form
- Higher Order Derivative
- Second Order Derivative

##### Applications of Derivative

##### Integration

- Methods of Integration - Integration by Substitution
- Methods of Integration - Integration Using Partial Fractions
- Methods of Integration - Integration by Parts
- Definite Integral as the Limit of a Sum
- Fundamental Theorem of Calculus
- Properties of Definite Integrals
- Evaluation of Definite Integrals by Substitution
- Integration by Non-repeated Quadratic Factors

##### Applications of Definite Integral

##### Differential Equation

- Basic Concepts of Differential Equation
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Formation of Differential Equation by Eliminating Arbitary Constant
- Differential Equations with Variables Separable Method
- Homogeneous Differential Equations
- Linear Differential Equation
- Applications of Differential Equation

##### Statistics

##### Probability Distribution

- Conditional Probability
- Random Variables and Its Probability Distributions
- Discrete and Continuous Random Variable
- Probability Mass Function (P.M.F.)
- Probability Distribution of a Discrete Random Variable
- Cumulative Probability Distribution of a Discrete Random Variable
- Expected Value, Variance and Standard Deviation of a Discrete Random Variable
- Probability Density Function (P.D.F.)
- Distribution Function of a Continuous Random Variable

##### Bernoulli Trials and Binomial Distribution

#### definition

If E and F are two events associated with the same sample space of a random experiment, the conditional probability of the event E given that F has occurred, i.e. P (E|F) is given by

P(E|F) = `(P(E ∩ F))/(P(F))` provided P(F) ≠ 0

#### notes

Consider the experiment of tossing three fair coins. The sample space of the experiment is

S = {HHH ,HHT,HTH , THH , HTT , THT , TTH , TTT }

Since the coins are fair, we can assign the probability `1/8` to each sample point. Let E be the event ‘at least two heads appear’ and

F be the event ‘first coin shows tail’. Then

E = {HHH, HHT, HTH, THH} and

F = {THH, THT, TTH, TTT}

Therefore P(E) = P ({HHH}) + P ({HHT}) + P ({HTH}) + P ({THH})

= `1/8 + 1/8 + 1/8 + 1/8 = 1/2 `

and P(F) = P ({THH}) + P ({THT}) + P ({TTH}) + P ({TTT})

= `1/8 + 1/8 + 1/8 + 1/8 = 1/2`

Also E ∩ F = {THH}

with P(E ∩ F) = P({THH}) = `1/8`

Now, the sample point of F which is favourable to event E is THH. Thus, Probability of E considering F as the sample space = `1/4`

or Probability of E given that the event F has occurred = `1/4`

This probability of the event E is called the conditional probability of E given that F has already occurred, and is denoted by P (E|F).

Thus P (E | F) = `1/4`

Note that the elements of F which favour the event E are the common elements of E and F, i.e. the sample points of E ∩ F.

Thus, we can also write the conditional probability of E given that F has occurred as

P (E|F) =

`("Number of elementary events favourable to E ∩ F")/("Number of elementary events which are favourable to F")`

`= (n(E ∩ F))/(n(F))`

Dividing the numerator and the denominator by total number of elementary events of the sample space, we see that P(E|F) can also be written as

P(E|F) =` (n(E ∩ F))/(n(S)) / (n(F))/(n(S)) = (P(E ∩ F))/(P(F))` ...(1)

Note that (1) is valid only when P(F) ≠ 0 i.e., F ≠ φ

#### Video Tutorials

#### Shaalaa.com | Probability part 11 (Example :- Conditional Probability)

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