Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Probability ↓


Introduction to Probability


Probability is a fascinating area of mathematics that deals with the likelihood or probability of events occurring. It helps us understand how likely events are to occur, from everyday events like predicting the weather to sports and even understanding complex systems. The concept of probability is not only central to statistics, but also applies in real-life decision making.

Basic concepts

Let's start with some basic concepts that you need to know to understand probability:

Use

An experiment is an action or process that leads to a set of results. For example, tossing a coin, throwing dice, or drawing a card from a deck are all experiments.

Outcome

An outcome is a possible result of an experiment. If you flip a coin, the possible outcomes are heads or tails.

Sample space

The sample space is the set of all possible outcomes of an experiment. It is usually denoted by the letter S. For a single toss of a coin, the sample space is:

S = {Heads, Tails}

Events

An event is a specific outcome or a set of outcomes. For example, getting heads when you flip a coin is an event. If you are throwing dice, getting an even number such as 2, 4, or 6 is another example of an event.

Calculating Probability

Probability allows us to calculate the likelihood of different events. It is expressed as a number between 0 and 1, where 0 means that the event will not happen, and 1 means that it will definitely happen. If P(E) is the probability of an event E, then it is calculated as follows:

P(E) = (Number of favorable outcomes) / (Total number of possible outcomes)

Example: tossing a coin

Let us consider the simple experiment of tossing a coin. What is the probability of getting heads?

  • Total number of possible outcomes = 2 (heads, tails)
  • Number of favourable outcomes = 1 (heads)
P(Heads) = 1 / 2 = 0.5

The probability of heads coming up is 0.5 or 50%.

Example: Rolling a dice

Consider a six-sided die. What is the probability of getting a four?

  • Total number of possible outcomes = 6 (1, 2, 3, 4, 5, 6)
  • Number of favourable outcomes = 1 (4)
P(rolling a 4) = 1 / 6 ≈ 0.1667

The probability of getting a four is approximately 0.1667 or 16.67%.

Types of events

Not all events are simple. Let us look at the different types of events based on probability:

Independent events

Two events are independent if the occurrence of one does not affect the occurrence of the other. For example, tossing a coin and throwing a dice are independent events. The outcome of the coin toss does not affect the roll of the dice.

Dependent events

Events are dependent if the outcome of the first event or events affects the outcome of the second event. If you draw a card from the deck and do not replace it, then draw another card, the events are dependent.

Mutually exclusive events

If events cannot occur at the same time then they are mutually exclusive. For example, when rolling a standard dice, the events of getting an odd number (1, 3, 5) and an even number (2, 4, 6) are mutually exclusive.

Complementary programs

The complement of an event E is the event that E does not occur. The sum of the probabilities of an event and its complement is 1.

Example including view: spinner

Imagine a spinner divided into four equal parts, named A, B, C and D. If we spin the spinner, what is the probability of landing on section A?

ABCD
  • Total number of possible outcomes = 4 (A, B, C, D)
  • Number of favourable outcomes = 1 (A)
P(A) = 1 / 4 = 0.25

The probability of landing on A is 0.25 or 25%.

Uses of probability in real life

Probability isn't just a number you calculate for games or academic purposes. It has many real-life applications, such as:

  • Weather forecasting: Meteorologists use probability to forecast the chance of rain or sunshine.
  • Insurance: Insurance companies assess risks and use probability to set premiums and cover costs.
  • Medical field: Probability is used in determining the success rate of treatments and outcomes.

Conclusion

Understanding probability helps us make better decisions by analyzing the likelihood of specific outcomes. As we have seen, probability can be applied to various elements of life, from simple games to complex, real-world predictive models. This introduction covers the basic principles, and as you progress, you will find more sophisticated techniques and theories to explore.


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Introduction to Probability

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