#### Topics

##### Number Systems

##### Real Numbers

##### Algebra

##### Polynomials

##### Pair of Linear Equations in Two Variables

- Linear Equation in Two Variables
- Graphical Method of Solution of a Pair of Linear Equations
- Substitution Method
- Elimination Method
- Cross - Multiplication Method
- Equations Reducible to a Pair of Linear Equations in Two Variables
- Consistency of Pair of Linear Equations
- Inconsistency of Pair of Linear Equations
- Algebraic Conditions for Number of Solutions
- Simple Situational Problems
- Pair of Linear Equations in Two Variables
- Relation Between Co-efficient

##### Quadratic Equations

- Quadratic Equations
- Solutions of Quadratic Equations by Factorization
- Solutions of Quadratic Equations by Completing the Square
- Nature of Roots of a Quadratic Equation
- Relationship Between Discriminant and Nature of Roots
- Situational Problems Based on Quadratic Equations Related to Day to Day Activities to Be Incorporated
- Application of Quadratic Equation

##### Arithmetic Progressions

##### Coordinate Geometry

##### Lines (In Two-dimensions)

##### Constructions

- Division of a Line Segment
- Construction of Tangents to a Circle
- Constructions Examples and Solutions

##### Geometry

##### Triangles

- Similar Figures
- Similarity of Triangles
- Basic Proportionality Theorem (Thales Theorem)
- Criteria for Similarity of Triangles
- Areas of Similar Triangles
- Right-angled Triangles and Pythagoras Property
- Similarity of Triangles
- Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
- Triangles Examples and Solutions
- Angle Bisector
- Similarity of Triangles
- Ratio of Sides of Triangle

##### Circles

- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Tangent to a Circle
- Number of Tangents from a Point on a Circle
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles

##### Trigonometry

##### Introduction to Trigonometry

- Trigonometry
- Trigonometry
- Trigonometric Ratios
- Trigonometric Ratios and Its Reciprocal
- Trigonometric Ratios of Some Special Angles
- Trigonometric Ratios of Complementary Angles
- Trigonometric Identities
- Proof of Existence
- Relationships Between the Ratios

##### Trigonometric Identities

##### Some Applications of Trigonometry

##### Mensuration

##### Areas Related to Circles

- Perimeter and Area of a Circle - A Review
- Areas of Sector and Segment of a Circle
- Areas of Combinations of Plane Figures
- Circumference of a Circle
- Area of Circle

##### Surface Areas and Volumes

- Surface Area of a Combination of Solids
- Volume of a Combination of Solids
- Conversion of Solid from One Shape to Another
- Frustum of a Cone
- Concept of Surface Area, Volume, and Capacity
- Surface Area and Volume of Different Combination of Solid Figures
- Surface Area and Volume of Three Dimensional Figures

##### Statistics and Probability

##### Statistics

##### Probability

##### Internal Assessment

- Classical Definition of Probability
- Type of Event - Impossible and Sure Or Certain
- assume that all the experiments have equally likely outcomes, impossible event, sure event or a certain event, complementary events,

## Notes

Probobility means chances or possibilities of happening something.

1) `"P(E)"= "No. of fovourable outcomes"/ "Total No. of possible outcomes"`

Where, P(E) is read as probability of an event, fovourable outcomes are the outcomes that we are asked find in the question, total no. of possible outcomes is the total possible result.

Example1 - `"P(H)"= "Probability of getting head"= 1/2`

`"P(T)"= "Probability of getting tail"= 1/2`

Example2 - `"P(S)"= "Probability of getting 5 on a die"= 1/6`

2) Impossible Event- Some events can never take place, they are known as impossible events. The probablity of impossible events is always 0.

Example1 - Probability of getting 8 on a die `"P(E)"= 0/6= 0`

Example2 - Probability of sun revolving around the earth P(E)= 0

3) Sure Event- This are events which will definetly happen. The probablity of sure events is always 1.

Example- Probability of getting less than 7 on a die

`"P(E)"= 6/6= 1`

4) Complementary Events- The sum of probablities of events that will happen and the one that will not happen is known as complementary events. The sum of complemetary event is always 1. P(E)+ P(not E)=1.

P(not E)= 1- P(E)

Example- Probability of getting 3 on a die `"P(E)"= 1/6`

`"P(E)"+ "P"("not E")= 1`

`1/6+ "P"("not E")= 1`

`"P"("not E")= 1- 1/6`

`"P"("not E")= 5/6`

5) Note-

i) Probability for any event- 0 greater than equal to P(E) greater that equal to 1.

ii) Probability can neither be negative nor be greater than one.

iii) An event having only one outcome of the experiment is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1.

iv) Sum of all the prbabilities of some events is one i.e

`P(E_1)+ P(E_2)+ P(E_3)+ .......= 1`

Example- `P(E_1)= 1/3, P(E_2)= 1/4, P(E_3)= ?`

Solution- `P(E_1)+ P(E_2)+ P(E_3)= 1`

`1/3+ 1/4+ P(E_3)= 1`

`P(E_3)= 1- 7/12`

`P(E_3)= 5/12`

6) Coins- In the case of coins we will toss upto three coins. Sample space means number of possible outcomes.

i) One coin- Sample space= Head(H), Tail(T). Total number of outcomes are 2.

ii) Two coins- Sample space= HH, HT, TH, HH. Total number of outcomes are 4.

iii) Three coins- Sample space= HHH, HHT, HTH, THH, TTT, TTH, THT, HTT. Total number of outcomes are 8.

7) Die- Here we will the sample space upto two dice.

i) One die- sample space= 1,2,3,4,5,6. Total number of outcomes are 6.

ii) Two dice- sample space= (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Total number of outcomes are 36

Doublet, means chances of getting same numbers on both the dice- sample space= (1,1) (2,2) (3,3) (4,4) (5,5) (6,6).

Total number of outcomes are 6.

8) Playing cards- The total numbers of cards in a pack are 52, and the number of suit i.e number of types are 4.

The 4 suits are spade, club, heart and diamond. There are only two colours in a pack of cards that are red and black. While spade and club are always black and heart and diamond are always red.

Out of total 52 cards 26 cards are red and other 26 are black. There are 13 cards of each suit. Eg. Spade will have cards like 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), K (king), A (ace), every suit will all this cards.

Face cards or picture cards are the cards which have face on it i.e like Jack, Queen and King. There are in total 12 face cards. Because every suit will have one jack, one queen and one king, that's why there will be 12 face cards.

And if it is asked that in a pack of cards how many red face cards we have? Then the answer wil be 6. There are 6 red face cards and 6 black face cards.

#### Video Tutorials

#### Shaalaa.com | Probability part 2 (Classical Approach)

#### Related QuestionsVIEW ALL [86]

A die has 6 faces marked by the given numbers as shown below:

1 | 2 | 3 | - 1 | - 2 | - 3 |

The die is thrown once. What is the probability of getting a positive integer?