Probability (Forms 5–6)

Arts 1–4

Foreign Languages 1–4

Covers probability rules, tree diagrams, and independent/dependent events.

📘 Topic Summary

Probability is a fundamental concept in Additional Mathematics that deals with the chance or likelihood of an event occurring. It's essential to understand probability rules, tree diagrams, and independent/dependent events to make informed decisions and solve problems effectively.

📖 Glossary

Event: A specific outcome or occurrence.
Sample Space: The set of all possible outcomes or results.
Probability Measure: A function that assigns a number between 0 and 1 to each event, representing its likelihood.
Independent Events: Events where the occurrence of one event does not affect the probability of another event.
Dependent Events: Events where the occurrence of one event affects the probability of another event.

⭐ Key Points

The probability of an event is a number between 0 and 1, inclusive.
The sum of the probabilities of all possible outcomes in a sample space is always 1.
Independent events can be combined using the multiplication rule: P(A ∩ B) = P(A) × P(B).
Dependent events require the use of conditional probability: P(A|B) = P(A ∩ B) / P(B).
Tree diagrams are a useful tool for visualizing and calculating probabilities in complex scenarios.
The probability of an event not occurring is 1 minus its probability: P(not A) = 1 - P(A).

🔍 Subtopics

Introduction to Probability

Probability is a branch of mathematics that deals with the chance or likelihood of an event occurring. It is used to describe and analyze random events, which are events that have more than one possible outcome. The probability of an event is usually denoted by P(A) and is measured on a scale from 0 (impossible) to 1 (certain).

Independent Events

Two events A and B are said to be independent if the occurrence or non-occurrence of one event does not affect the probability of the other event. The probability of both events occurring is calculated by multiplying their individual probabilities: P(A ∩ B) = P(A) × P(B).

Dependent Events

Two events A and B are said to be dependent if the occurrence or non-occurrence of one event affects the probability of the other event. The probability of both events occurring is calculated using a tree diagram, which shows all possible outcomes and their corresponding probabilities.

Bayes' Theorem

Bayes' theorem is a mathematical formula used to update the probability of an event based on new information or evidence. It states that P(A|B) = P(B|A) × P(A) / P(B), where A and B are events.

Random Variables

A random variable is a variable whose possible values are determined by chance, rather than by a fixed set of rules. Random variables can be discrete or continuous, depending on the number of possible outcomes.

Probability Distributions

A probability distribution is a function that describes the probability of each possible value of a random variable. Common examples include the uniform distribution and the normal distribution.

Chebyshev's Inequality

Chebyshev's inequality states that for any random variable X, at least (1 - 1/k^2) proportion of the values will lie within k standard deviations from the mean. This is a useful tool for bounding the probability of extreme events.

Markov Chains

A Markov chain is a mathematical system that undergoes transitions from one state to another according to certain rules. The transition probabilities are used to predict the long-term behavior of the system.

Random Processes

A random process is a sequence of random variables indexed by time or space. Random processes can be used to model real-world phenomena, such as stock prices or weather patterns.

Simulation and Modeling

Simulation and modeling are techniques used to approximate the behavior of a system or process using probability theory. This is useful for predicting outcomes in complex systems where exact calculations are difficult or impossible.

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