Day 2: The Bedrock of Bayes' Theorem

1. The Two Unbreakable Laws

1. Conditional Probability Law – The chance of A happening and B happening is the chance of B happening times the chance of A happening given B.
   Analogy: Imagine a bag of red and blue marbles. If you first pick a blue marble (B), the chance you pick a red one next (A) depends on how many red marbles are left.
2. Law of Total Probability – The chance of A happening is the sum of the chances of A happening in each possible situation (B).
   Analogy: Think of a pizza cut into slices. The chance of picking a slice with pepperoni (A) is the sum of the chances of picking pepperoni from each slice type (B).

FIRST PRINCIPLE:
“The probability of an event equals the probability of the event’s cause times the probability of the cause, summed over all possible causes.”

2. Bayes' Theorem – Inverting the Direction

Bayes' theorem flips the Conditional Probability Law.

• Formula:
  [ P(A|B) = \frac{P(B|A) \times P(A)}{P(B)} ]
• What it does:
  It lets you find the chance of a cause (A) when you only know the effect (B).
• Real‑world use:
  Doctors use it to update the chance a patient has a disease after a test result.