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


Probability is a part of mathematics that deals with the likelihood or probability of various outcomes. It is used to estimate how likely events are to occur. Probability helps us understand various everyday situations and make informed decisions. It is an important concept not only in mathematics but also in fields such as statistics, science, and finance.

Understanding probability

The concept of probability is basically about measuring uncertainty. When we talk about probability, we are often interested in the likelihood of a particular event occurring. To describe probability in mathematical terms, we typically use numbers from 0 to 1:

  • A probability of 0 means the event will not occur.
  • Probability 1 means that the event will definitely happen.
  • Probability between 0 and 1 indicates different levels of probability. For example, a probability of 0.5 indicates that an event is equally likely to occur or not occur.

The basic formula of probability is:

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

Examples of simple probability

Let's look at a basic example to understand this concept better:

Example 1: Tossing a coin

Consider a standard coin that has two sides: heads and tails. If we flip this coin, what is the probability that heads occur?

There is one favorable desired outcome (heads) and two possible outcomes (heads and tails). Therefore, the probability of getting heads is:

P(head) = 1 / 2 = 0.5
Head Tail

Example 2: Throwing a dice

Consider a standard six-sided die with numbers 1 to 6 marked on its edges. If we throw this die, what is the probability of getting a 4?

Of the six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6), there is one favorable outcome (rolling a 4). Thus, the probability of rolling a 4 is:

P(rolling a 4) = 1 / 6 ≈ 0.1667
1, 2, 3, 4 , 5, 6

Complementary probability

Complementary probability is the probability of an event not occurring. It is calculated by subtracting the probability of the event from 1. This is another way of looking at the probability of events.

Example 3: 4's complement

What is the probability that a six-sided dice does not show 4?

Given that the probability of getting 4 is 0.1667, the probability of not getting 4 is:

P(4 not coming) = 1 - P(4 coming) = 1 - 1/6 = 5/6 ≈ 0.8333

Types of probability

There are several different ways to understand and calculate probability. These include theoretical probability, experimental probability, and subjective probability.

Theoretical probability

Theoretical probability is based on the expected chances of an outcome. This probability is calculated using this formula:

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

We have already seen examples of theoretical probability in tossing a coin and throwing a dice.

Experimental probability

Experimental probability is based on direct observation or experiments. It is calculated by dividing the number of times an event occurs by the total number of trials.

Example 4: Tossing a coin 100 times

If you toss a coin 100 times and get heads 55 times, then the experimental probability of getting heads is:

P(heads) = Number of heads / Total number of tosses = 55 / 100 = 0.55

Experimental probability may differ from theoretical probability, as it is based on actual trials, which may not perfectly align with expected outcomes due to randomness.

Subjective probability

Subjective probability is based on personal reasoning or opinion rather than precise data. For example, a weather forecast stating that there is a 70% chance of rain tomorrow is based on subjective probability informed by various meteorological models and expertise.

Visualizing probability with events

Often, it is helpful to see the probabilities visually to understand the relationships between different events.

All possible outcomes events

In this diagram, the large rectangle represents the entire sample space (all possible outcomes), while the smaller shaded area represents a specific event within that sample space.

Mixed events

Mixed events involve the possibility of two or more outcomes occurring simultaneously. There are various rules and formulas for determining probabilities in these situations.

Independent events

Independent events do not affect each other's outcomes. The probability of several independent events occurring is the product of their individual probabilities.

Example 5: Independent events

If you throw two dice what is the probability that both will come up as 6?

The probability of getting 6 on a die is 1/6. Therefore, the probability of getting 6 on both dice is:

P(die 1 = 6 and die 2 = 6) = 1/6 * 1/6 = 1/36 ≈ 0.0278

Dependent events

Dependent events occur when the outcome of one event affects the outcome of another event. Calculating probabilities associated with dependent events requires adjusting probabilities based on the outcomes of previous events.

Example 6: Dependent events

Suppose you have a bag containing 3 red and 2 blue marbles. If you draw out two marbles without replacing the first one, what is the probability that both are red?

Probability that the first marble is red:

P(first red) = 3/5

The display of the second draw will now be affected:

The probability that the second marble is red after the first marble is red:

P(second red | first red) = 2/4 = 1/2

Thus, the joint probability is:

P(both red) = P(1st red) * P(2nd red | 1st red) = 3/5 * 1/2 = 3/10 = 0.3

Additional examples

Now, let us look at some more examples to make the concept of probability more clear:

Example 7: Creating a card

From a standard 52-card deck, what is the probability of drawing a queen?

There are 4 queens in a deck of 52 cards. Therefore, the probability of drawing a queen is:

P(queen) = 4 / 52 = 1 / 13 ≈ 0.0769

Example 8: At least one event occurs

What is the probability that when two dice are thrown at least one dice will be a 6?

To calculate this probability, it is often easier to use the complement. First, calculate the probability of not rolling a 6 on any of the dice:

Probability of not getting 6 on a dice:

P(not 6) = 5/6

Probability of not getting 6 on both dice:

P(not 6 on both) = (5/6) * (5/6) = 25/36

Probability of getting at least one 6:

P(at least one 6) = 1 - P(not both 6) = 1 - 25/36 = 11/36 ≈ 0.3056

Final thoughts

Probability is an important part of mathematics that enables us to estimate how likely events are to occur, whether in simple cases like throwing dice or in more complex scenarios involving multiple events. Having a solid understanding of probability helps improve decision-making skills in uncertain situations, making it invaluable in many fields, including science, engineering, finance, and everyday life.

As you continue to explore probability, remember that practice and real-world application are the keys to mastering these concepts. Continue working through a variety of examples and experiments to become familiar with the many facets of probability.


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