Numerical Systems

Introduction Positional Notation Converting Between Bases Binary Addition Binary Multiplication Challenges (Python) Challenges (Java)

Modular Arithmatic

Introduction The Modulus Operator Cyphers Modular Arithmetic Modular Inverse Challenges (Java)

Algorithms

Introduction Algorithms Time Functions Asymptotic Complexity Summation Challenges

Propositional Logic

Introduction Propositional Logic Conjunction and Disjunction Implication and Equivalence Bitwise Operators Challenges

Set Theory

Introduction Sets Subsets and Supersets Set Operations Challenges

Graph Theory

Introduction Graphs Directed Graphs Structure of Language Search Trees Neural networks Game Object Trees Challenges Graph Question Generator

Proof

Introduction Proof Proof by Cases Proof by Contradiction Proof by Induction Challenges

Maths to Code

Maths to Code Challenges Question Generator

Descriptive Statistics

Introduction Statistics Measures of Central Tendency Measures of Spread Data Visualisation Correlation Challenges

Probability

Introduction Probability Basics Combinatorics Probability Trees Law of Total Probability Bayes Rule Challenges

Inferential Statistics

Introduction Probability Distributions Samples and Populations Hypothesis Testing Z Scores Student's T-Test Challenges

Law of Total Probability

In this section we’re going to discover the Law of Total Probability.

Watch the video and then answer the questions below.

Ten-minute video

You can also view this video on YouTube

Key Points

Partition

A partition is a way of subdividing a set into partitions \( B_1 \), \( B_2 \), …, \( B_n \) that are:

disjoint (i.e. there is no overal between \( B_i \) and \( B_j \), or in other words \( B_i \cap Bj = \emptyset \))
collectively exhaustive (i.e. together they contain everything in the set)

We can often partition our sample space. For example, if we have an experiment \( S \) where we pick a ball from a random bag out of three bags, we can draw from either \( B_1 \), \( B_2 \), or \( B_3 \). As there are no other outcomes in our experiment, these would be partitions of our sample space.

Any other event in the sample space \( A \) (e.g. the ball being red) would overlap one or more of these partitions. Then we can work out \( P(A) \) as follows: