Class 11 Mathematics (041) · Chapter 14 NotesProbability
Learn about probability, including events, sample space, axioms, and applications. Understand how to calculate probabilities and apply them to real-world problems.
Probability is a fundamental concept in mathematics that deals with the likelihood of events occurring. In this chapter, we explore the axiomatic approach to probability, which provides a rigorous foundation for understanding and calculating probabilities. We begin by defining key terms such as events, sample space, and outcomes. We then introduce the axioms of probability, which include the non-negativity, normalization, and additivity of probabilities. These axioms form the basis for all probability calculations. The chapter also covers important concepts such as the probability of an event, the probability of equally likely outcomes, and the probability of the event 'A or B'. Through examples and exercises, we learn how to apply these concepts to solve real-world problems. By the end of the chapter, students will have a solid understanding of probability and its applications, which is essential for further studies in statistics and other areas of mathematics.
What you'll learn
1Understand the basic concepts of probability, including events, sample space, and outcomes
2Learn the axioms of probability and their implications
3Calculate the probability of an event using the axiomatic approach
4Determine the probability of equally likely outcomes
5Apply the probability of the event 'A or B' to solve problems
6Solve real-world problems using probability concepts
Detailed chapter notes
01
Introduction to Probability
Probability is a measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The sample space of an experiment is the set of all possible outcomes. An event is a subset of the sample space. For example, when tossing a coin, the sample space is {Heads, Tails}, and an event could be 'the coin lands on Heads'. The probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes. In the case of the coin toss, the probability of landing on Heads is 1/2.
Probability measures the likelihood of an event occurring
Sample space is the set of all possible outcomes
An event is a subset of the sample space
Probability is calculated as the ratio of favorable outcomes to total outcomes
02
Axioms of Probability
The axiomatic approach to probability provides a formal framework for understanding and calculating probabilities. The three axioms of probability are: (1) The probability of any event is non-negative, (2) The probability of the sample space is 1, and (3) The probability of the union of two mutually exclusive events is the sum of their individual probabilities. These axioms ensure that probability calculations are consistent and reliable. For example, if we have two mutually exclusive events A and B, the probability of A or B occurring is P(A) + P(B).
Three axioms of probabilitynon-negativity, normalization, and additivity
Axioms ensure consistency and reliability in probability calculations
Probability of the union of two mutually exclusive events is the sum of their probabilities
03
Probability of an Event
The probability of an event is the measure of the likelihood that the event will occur. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. For example, if we roll a die, the probability of rolling a 3 is 1/6, since there is one favorable outcome (rolling a 3) and six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6). The probability of an event can also be calculated using the axiomatic approach, where probabilities are assigned to each outcome in the sample space, and the probability of an event is the sum of the probabilities of the outcomes in the event.
Probability of an event is the ratio of favorable outcomes to total outcomes
ExampleProbability of rolling a 3 on a die is 1/6
Probability can be calculated using the axiomatic approach
04
Probability of Equally Likely Outcomes
When all outcomes in the sample space are equally likely, the probability of each outcome is the same. For example, when tossing a fair coin, the probability of landing on Heads or Tails is 1/2. The probability of an event in this case is the number of favorable outcomes divided by the total number of possible outcomes. For example, the probability of rolling an even number on a die is 3/6, since there are three favorable outcomes (2, 4, 6) and six possible outcomes. This concept is useful in many real-world applications, such as games of chance and statistical analysis.
Equally likely outcomes have the same probability
Probability of an event is the ratio of favorable outcomes to total outcomes
ExampleProbability of rolling an even number on a die is 3/6
05
Probability of the Event 'A or B'
The probability of the event 'A or B' is the probability that either A or B or both will occur. It is calculated as the sum of the probabilities of A and B minus the probability of both A and B occurring. This is known as the inclusion-exclusion principle. For example, if we have two events A and B, the probability of A or B occurring is P(A) + P(B) - P(A and B). This concept is useful in many real-world applications, such as calculating the probability of multiple events occurring in a sequence or the probability of at least one event occurring in a set of events.
Probability of 'A or B' is P(A) + P(B) - P(A and B)
Inclusion-exclusion principle is used to calculate the probability of the union of two events
ExampleProbability of A or B occurring is P(A) + P(B) - P(A and B)
06
Applications of Probability
Probability has numerous applications in various fields, including statistics, finance, engineering, and science. In statistics, probability is used to calculate the likelihood of different outcomes in experiments and surveys. In finance, probability is used to assess the risk of investments and to make decisions about portfolio management. In engineering, probability is used to design and analyze systems that are subject to random variations. In science, probability is used to model and predict the behavior of complex systems. By understanding and applying the concepts of probability, we can make informed decisions and solve real-world problems.
Probability is used in statistics, finance, engineering, and science
In statistics, probability is used to calculate the likelihood of outcomes
In finance, probability is used to assess risk and make investment decisions
In engineering, probability is used to design and analyze systems
In science, probability is used to model and predict complex systems
Want the complete chapter resources?Topic notes, quizzes and flashcards for Probability.
What is the difference between an event and a sample space?
A sample space is the set of all possible outcomes of an experiment, while an event is a subset of the sample space that corresponds to a specific outcome or set of outcomes.
How do you calculate the probability of an event?
The probability of an event is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
What are the three axioms of probability?
The three axioms of probability are: (1) The probability of any event is non-negative, (2) The probability of the sample space is 1, and (3) The probability of the union of two mutually exclusive events is the sum of their individual probabilities.
What is the probability of the event 'A or B'?
The probability of the event 'A or B' is calculated as P(A) + P(B) - P(A and B). This is known as the inclusion-exclusion principle.
What are some real-world applications of probability?
Probability has numerous real-world applications, including in statistics, finance, engineering, and science. It is used to calculate the likelihood of outcomes, assess risk, design systems, and model complex behaviors.