Probability Rules Overview
Author: Mcenroe Ng
Content Page
1. Introduction
2. Bayes' Rule
2.1 Definition
2.2 Formula
2.3 Example
3. Multiplication Rule
3.1 Definition
3.2 Formula
3.3 3.3 Example
4. Law of Total Probability
4.1 Definition
4.2 Formula
4.3 Example
5. Addition Rule
5.1 Definition
5.2 Formula
5.3 Example
6. Comparison of Rules and Scenarios
7. Conclusion
, 1. Introduction Probability is a branch of mathematics that deals with quantifying uncertainty and
measuring the likelihood of events. In probability theory, there are several important rules and
formulas that help calculate probabilities in different situations.
This note explores four key concepts: Bayes' Rule, the Multiplication Rule, the Law of Total
Probability, and the Addition Rule. Each formula is described, along with examples and guidance on
when to use them. Additionally, a comparison table is provided to help determine which rule to
apply in different scenarios.
2. Bayes' Rule
2.1 Definition: Bayes' Rule, also known as Bayes' Theorem or Bayes' Law, is used to update the
probability of an event based on new information or evidence.
2.2 Formula:
The formula for Bayes' Rule is as follows: P(A|B) = (P(B|A) * P(A)) / P(B)
Where:
• P(A|B) is the probability of event A given event B has occurred.
• P(B|A) is the probability of event B given event A has occurred.
• P(A) is the prior probability of event A.
• P(B) is the prior probability of event B.
2.3 Example:
Suppose there is a bag of red and blue balls. The bag contains 6 red balls and 4 blue balls. You
randomly pick a ball without looking and it is red. What is the probability that the next ball you pick
will also be red?
Solution:
Let A be the event "the second ball is red," and B be the event "the first ball is red." Using Bayes'
Rule: P(A|B) = (P(B|A) * P(A)) / P(B) P(A|B) = (5/9 * 6/10) / (6/10) P(A|B) = 5/9
Therefore, the probability that the second ball will be red, given that the first ball was red, is 5/9.
Author: Mcenroe Ng
Content Page
1. Introduction
2. Bayes' Rule
2.1 Definition
2.2 Formula
2.3 Example
3. Multiplication Rule
3.1 Definition
3.2 Formula
3.3 3.3 Example
4. Law of Total Probability
4.1 Definition
4.2 Formula
4.3 Example
5. Addition Rule
5.1 Definition
5.2 Formula
5.3 Example
6. Comparison of Rules and Scenarios
7. Conclusion
, 1. Introduction Probability is a branch of mathematics that deals with quantifying uncertainty and
measuring the likelihood of events. In probability theory, there are several important rules and
formulas that help calculate probabilities in different situations.
This note explores four key concepts: Bayes' Rule, the Multiplication Rule, the Law of Total
Probability, and the Addition Rule. Each formula is described, along with examples and guidance on
when to use them. Additionally, a comparison table is provided to help determine which rule to
apply in different scenarios.
2. Bayes' Rule
2.1 Definition: Bayes' Rule, also known as Bayes' Theorem or Bayes' Law, is used to update the
probability of an event based on new information or evidence.
2.2 Formula:
The formula for Bayes' Rule is as follows: P(A|B) = (P(B|A) * P(A)) / P(B)
Where:
• P(A|B) is the probability of event A given event B has occurred.
• P(B|A) is the probability of event B given event A has occurred.
• P(A) is the prior probability of event A.
• P(B) is the prior probability of event B.
2.3 Example:
Suppose there is a bag of red and blue balls. The bag contains 6 red balls and 4 blue balls. You
randomly pick a ball without looking and it is red. What is the probability that the next ball you pick
will also be red?
Solution:
Let A be the event "the second ball is red," and B be the event "the first ball is red." Using Bayes'
Rule: P(A|B) = (P(B|A) * P(A)) / P(B) P(A|B) = (5/9 * 6/10) / (6/10) P(A|B) = 5/9
Therefore, the probability that the second ball will be red, given that the first ball was red, is 5/9.
