Probability Trees and Bayes' Theorem

Probability Trees

A probability tree diagram shows all the possible events or outcomes involving one or more probabilistic events. Each branch of the tree represents a possible outcome and has a probability associated with it. The outcomes at each level of the tree are mutually exclusive, and the probabilities of all outcomes at each level must add up to 1.

A probability tree can be used to calculate the probability of an event by multiplying the probabilities of the branches that lead to the event.

Bayes' Theorem

Bayes' theorem, named after Thomas Bayes, describes the probability of an event based on prior knowledge of conditions that might be related to the event. It serves as a way to figure out conditional probability.

Given a hypothesis H and evidence E, Bayes' theorem states that the relationship between the probability of the hypothesis before getting the evidence P(H) and the probability of the hypothesis after getting the evidence P(H|E) is:

P(H|E) = [P(E|H) * P(H)] / P(E)

Where:

• P(H|E) is the probability of the hypothesis H given the evidence E. This is called the posterior probability.
• P(E|H) is the probability of the evidence given that the hypothesis is true.
• P(H) and P(E) are the probabilities of the hypothesis and the evidence respectively.

Bayes' theorem is fundamental to Bayesian statistics, and has applications in a wide range of fields including science, engineering, medicine, and law.