Conditional Probability, Dependence, and Independence
John M. Shea · 2024
This chapter covers several important topics in Probability, including conditional probability and statistical independence. It starts by introducing a couple of classic probability problems that can be solved using conditional probability, including a problem about drawing balls from a box and the Monty Hall Problem, which is about making choices as a contestant on a particular game show. The first main sections shows how to use simulation to estimate conditional probabilities. Example of how to calculate conditional probabilities for small sample spaces with equally likely outcomes are considered. A formal definition of conditional probability is given, and it is shown that given a probability space with probability measure P, a conditional probability measure derived from P satisfies the Axioms of Probability on that sample space. Examples are used to show how conditioning can change probabilities. Statistical independence is introduced using conditional probabilities to motivate the formal definition. Then conditional independence is introduced. Finally, chain rules and the Law of Total Probability are derived and explained. Multiple examples are used to illustrate these concepts.