Introduction

Relevance of the Topic

Probability of Complementary Events is a fundamental concept within the discipline of mathematics, specifically in Probability Theory. This theme is a logical construction that allows the analysis of possibilities in various contexts, from weather forecasting to the probability of winning the lottery. Moreover, understanding complementary events is a prerequisite for more advanced topics, such as the Law of Compound Probabilities.

Contextualization

Within the mathematics curriculum, the probability of complementary events is usually introduced in the 8th grade of Elementary School. This content is an extension of the study of independent and dependent events, and opens doors to understanding probability in a broader sense. Understanding the complement of an event and how it relates to its probability is essential for mathematical analysis and the development of students' logical thinking. Therefore, this theme is the starting point for building the probability mathematics foundation for students.

Theoretical Development

Components

• Sample Space (Ω): The set of all possible outcomes of an experiment is called the sample space. Each element of this set is called a sample point. For example, in rolling a die, the sample space is {1, 2, 3, 4, 5, 6}.

• Event (E): An event is a subset of the sample space. It represents an outcome (or set of outcomes) of an experiment. In other words, it is a collection of sample points. For example, in rolling a die, the event 'getting an even number' is represented by the set {2, 4, 6}.

• Complementary Event (E'): The complementary event of an event E, denoted by E', is the event that contains all sample points from the sample space that are not in E. In other words, it is the event that occurs when event E does not occur. If E is the event 'getting an even number' in rolling a die, then the complementary event E' is 'getting an odd number', represented by the set {1, 3, 5}.

Key Terms

• Probability of an Event (P(E)): The probability of an event E occurring is the ratio of the number of favorable outcomes to E and the total number of possible outcomes in the experiment. It ranges from 0 (impossible event) to 1 (certain event).

• Complement Probability Law: The probability of the complementary event of E is equal to 1 minus the probability of E. In other words, P(E') = 1 - P(E).

Examples and Cases

• Example 1 - Die Roll: In rolling a die, consider event E as 'getting an even number' and E' as 'getting an odd number'. Since there are three odd numbers and three even numbers on a six-sided die, the probabilities of E and E' are 0.5 or 50%, respectively.

• Example 2 - Deck of Cards: Consider a deck of 52 cards. If E is the event 'drawing a heart card' and E' is the event 'drawing a non-heart card', then the probability of E is 13/52 = 1/4, and the probability of E' is 39/52 = 3/4.

• Example 3 - Rain Chances: If the weather forecast indicates an 80% chance of rain, then the probability of no rain (complementary event) is 20%.

Detailed Summary

Key Points

• Understanding Events: Understanding that an event is a subset of the sample space (Ω), meaning a collection of possible outcomes of an experiment. In rolling a die, for example, the event 'getting an even number' is a subset of the sample space {1, 2, 3, 4, 5, 6}.

• Deciphering the Sample Space: The sample space is the set of all possible outcomes of an experiment. In the case of rolling a die, the sample space is given by the numbers from 1 to 6.

• Benefiting from Complements: The complementary event (E') contains the sample points from the sample space that are not in event E. In rolling a die, for example, if E is the event 'getting an even number', E' is the event 'getting an odd number'.

• Relationship between Probability and Complementary Events: The probability of the complementary event of E (P(E')) is equal to 1 - the probability of E (P(E)). In other words, the probability of getting an odd number in rolling a die is 1 - the probability of getting an even number.

Conclusions

• Versatility of Complements: Understanding and effectively using complementary events allows for a more comprehensive and accurate analysis of the probability of occurrence of subsets of the sample space.

• Calculation Strategy: The complement law enables the calculation of event probabilities using the probabilities of their complements. This strategy can be useful for solving complex problems where the direct probability of an event may not be easily calculable.

Suggested Exercises

1. Card Game: Consider a deck of 52 cards. What is the probability of drawing a non-spade card? (Hint: Use the probability of the complementary event).

2. Coin Toss: If we toss a fair coin, what is the probability of not getting heads? (Hint: Use the probability of the complementary event).

3. Balls in an Urn: 10 balls, numbered from 1 to 10, are placed in an urn. If we draw a ball at random, what is the probability of it not being ball number 7? (Hint: Use the probability of the complementary event).