Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Probability ↓


Simple Problems on Probability


Probability is an interesting and fundamental topic in mathematics that gives us a way to deal with uncertainty and possibility. It is essentially a measure of how likely an event is to occur. This concept is used in a variety of fields, from weather forecasting to decision making in business, and even in sports and daily life situations. Here, we will explore simple problems on probability and provide extensive examples to ensure understanding.

Understanding probability

Probability represents the likelihood of an event occurring and is a value between 0 and 1. If the probability of an event is 0, it means the event will not occur. If the probability is 1, it means the event is certain to occur. More common probabilities are fractions, decimals, or percentages between these two extremes.

The fundamental formula of probability is given as:

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

Example 1: Throwing a dice

Consider a fair six-sided die. When it is rolled, the die can fall on any one of the six faces. If we want to find the probability of rolling a “3”, there is only one favorable outcome (rolling up “3”) and six possible outcomes (rolling up any one of the six numbers).

Probability = 1/6

So the probability of getting "3" is 1/6.

Example 2: Tossing a coin

When you flip a coin, there are two possible outcomes - heads or tails. If you want to know the probability of heads, there is one favorable outcome and two possible outcomes.

Probability = 1/2

Visualization of probability

Visual examples can help to understand probability. Let us illustrate the example of tossing a coin with a simple diagram showing the sample space:

H Tea

Here, the circles marked "H" and "T" represent head and tail, respectively.

Mixed events

Sometimes, we encounter situations involving two or more simple events. These are called compound events. For example, when you throw two dice or toss two coins at the same time, these are compound events. The basic rules of probability can be applied to these scenarios as well.

Example 3: Throwing two dice

Let's consider throwing two six-sided dice. If you want to find the probability that the sum of the numbers on the dice is "7", you must first identify all the possible outcomes. There are 36 possible outcomes when you throw two dice.

The outcomes where the sum is 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). There are 6 such favorable outcomes.

Probability = 6/36 = 1/6

Probability of multiple events

When dealing with multiple events, the probability of both events occurring is found by multiplying the probabilities of each event, assuming the events are independent. For example, the probability of tossing a coin and getting heads, and then throwing a die and getting 4, is calculated as follows:

Probability = (1/2) * (1/6) = 1/12

Practice Problems

Problem 1: Removing the card

A standard deck of cards contains 52 cards, divided into four suits: hearts, diamonds, clubs and spades. Each suit has 13 cards. What is the probability of drawing a queen?

There are 4 queens in a deck of 52 cards.

Probability = 4/52 = 1/13

Problem 2: Tossing three coins

If you toss three coins, what is the probability that exactly two of them will land on heads?

When you toss three coins the possible outcomes are as follows:

  • HHH
  • HHT
  • HTH
  • THH
  • TTH
  • THT
  • HTT
  • TTT

There are 8 possible outcomes.

Three outcomes have exactly two heads: HHT, HTH, and THH.

Probability = 3/8

Conclusion

Probability means forecasting the likelihood of future events. Understanding the basic principles of probability and simple problems can greatly enhance your ability to predict outcomes in real life. Even though the above explanation has only scratched the surface of the vast subject of probability, it should serve as a basic guide to reasoning and solving simple probability problems.

Stay curious and keep exploring more challenging problems and concepts to deepen your understanding of probability. Good luck on your mathematical journey!


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