Day 1: Foundations of Bayes’ Theorem

1. The Bedrock of Probability

Probability is a number that tells how likely something is.

• Rule 1: Every probability is a number between 0 (impossible) and 1 (certain).
• Rule 2: The probability of an event happening given something else is the chance they happen together divided by the chance the “something else” happens.
  • Think of a pizza: the chance of picking a pepperoni slice if you already know you’re picking a slice is the number of pepperoni slices divided by the total number of slices.

FIRST PRINCIPLE:
“The probability of event A happening when B has already happened is the chance that A and B both happen, divided by the chance that B happens.”

2. Turning Cause and Effect Around

Bayes’ theorem lets us flip that idea.
If we know how often a test is positive when someone really has the disease (P(Positive|Disease)) and how often the disease occurs in the population (P(Disease)), we can find how likely someone actually has the disease when they test positive (P(Disease|Positive)).
It’s like a detective who knows how often a clue appears when a suspect is guilty, and uses that to guess if a suspect is guilty when the clue shows up.