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 ↓


Complementary Events


In the world of probability, one of the fundamental concepts is to understand the different types of events. Among these, complementary events are an important cornerstone. In this lengthy discussion, we will delve deep into what complementary events are, how they work, and why they are important in the grand scheme of probability. Our aim is to simplify the explanation so that students can easily understand the concept and apply it to different problems.

Understanding events in probability

Before exploring complementary events, it's important to understand what an "event" means in probability. An event is any specific outcome or set of outcomes for a random experiment. Think of an event as a subset of the sample space, which is the set of all possible outcomes of that experiment.

For example, consider a fair six-sided die. The sample space for a single roll includes these numbers: {1, 2, 3, 4, 5, 6}. An example of an event might be rolling an even number, which includes the numbers {2, 4, 6}.

What are complementary events?

Now that we know what events are, we can talk about complementary events. Complementary events are a pair of events where one event occurs only if the other event does not occur. In simple terms, they are mutually exclusive. This means that the two events cannot occur at the same time, and between them, they cover all possible outcomes of the experiment.

If we denote an event by A, then the complement of event A is usually denoted by A' or sometimes A^c. The key concept here is that if A is the event that causes something to happen, then A' is the event that causes this something not to happen.

Mathematical representation

The probability of an event A and its complement A' can be mathematically summarized as follows:

P(A) + P(A') = 1

This equation states that the probability of event A occurring and the probability of event A not occurring equals 1, or 100%. This is because considering both outcomes (event A occurring and event A not occurring) covers the entire probability space.

Visualizing complementary events

To understand this in a better way, let’s look at some diagrams. Imagine a circle that represents the sample space, and within it, a part of the circle represents event A. The remaining part of the circle represents the complement of A:

A' A

Examples of complementary events

Continuing with our examples, let's take a look at some scenarios where complementary events make it easier to understand probability:

Example 1: Tossing a coin

One of the simplest examples of complementary events is tossing a coin. When a coin is tossed, there are two possible outcomes: heads (H) or tails (T).

  • Suppose event A results in heads. Then A = {H}.
  • The complement of event A is not getting heads, which means getting tails. So A' = {T}.

The probabilities can be calculated as follows:

P(A) = 0.5
P(A') = 1 - P(A) = 0.5

Example 2: Single die roll

Consider your six-sided die once again. What happens if a number less than four comes up?

  • Suppose the event B is that a number less than four occurs. Then B = {1, 2, 3}.
  • The complement of event B (denoted as B') is to roll a number greater than or equal to four. So B' = {4, 5, 6}.

The probabilities can be calculated as follows:

P(B) = 3/6 = 0.5
P(B') = 1 - P(B) = 3/6 = 0.5

Applying complementary events in real-life scenarios

Understanding complementary events also helps in understanding real-world scenarios where probability predictions are involved. Let us consider some real-life examples:

Example 3: Weather forecast

Weather forecasts are often expressed in terms of probabilities. For example, a weather report might say there is a 70% chance of rain tomorrow.

  • Suppose event C is that it will rain tomorrow. Then, P(C) = 0.7.
  • Then, the complementary event C' is that it will not rain tomorrow. So, P(C') = 1 - 0.7 = 0.3.

Example 4: Quality control

Suppose a factory has a quality control system to check products for defects, and it finds that 5% of its products are defective.

  • Let the event D be that the product is defective. Then, P(D) = 0.05.
  • The complementary event D' is that the product is not defective. Therefore, P(D') = 1 - 0.05 = 0.95.

Solving problems using complementary events

Understanding complementary events can also make it easier to solve probability problems. By calculating the complement of an event, sometimes we can reach the answer more quickly or easily. Consider an example where this approach might be useful:

Example 5: Drawing a card from the deck

Suppose you draw a card from a standard deck of 52 cards. If you draw two cards without replacement, what is the probability of drawing at least one heart?

  • It might be easier to calculate the probability of the complementary event, which isn't heart-pulling at all.
  • The probability that the first card is not a heart is P(first card not a heart) = 39/52.
  • If the first card is not a heart, there are 51 cards left, and 38 of them are not hearts. So,
  • P(second card not a heart | first not a heart) = 38/51.

Thus, the probability of not drawing any hearts is:

P(no hearts drawn) = (39/52) * (38/51)

Then, the probability of the complement, that is drawing at least one heart, is:

P(at least one heart) = 1 - P(no hearts drawn)

Exercises for practice

To strengthen your understanding of complementary events, try solving these additional exercises. The answers are given at the end to check your work:

  1. A bag contains 10 red balls and 5 blue balls. What is the probability of drawing a red ball?
  2. If a coin is tossed three times, what is the probability that it will come up heads at least once?
  3. A student guesses all the answers to a 4-question true or false quiz. What is the probability that all the answers are correct?

Conclusion

Understanding complementary events is crucial to mastering probability concepts. By recognizing that the sum of the probability of an event and its complement is one, we can tackle many probability challenges more easily. Complementary events simplify the process of calculating probability in many scenarios, both theoretically and in the real world.

The bottom line is this: become comfortable identifying complementary events, and use this concept as a tool to solve probability problems efficiently and effectively.

Answers to exercises

  1. The probability of drawing a red ball is: P(Red) = 10/15 = 2/3.
  2. There are 8 possible outcomes when a coin is tossed three times. Only 1 of these outcomes is all tails. Therefore, P(At least one heads) = 1 - 1/8 = 7/8.
  3. The probability of guessing all answers correctly is (1/2)^4 = 1/16.

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