Introduction

Conditional Probability: Core Concept

Conditional probability allows you to update the probability of an event A given that another event B has occurred.
It answers: How does knowing that B happened affect the likelihood of A?

Written as: \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\)
This is the probability that both A and B happen, divided by the probability that B happens.
Conceptually: zoom into the space where B is true and ask how likely A is inside that space.

Imagine the total probability space as a big set Ω.
Event A is a subset of Ω; event B is another subset.
When B happens, we restrict our view to only the region B.
Then P(A∣B)P(A \mid B) is the ratio of the overlapping part of A and B to the total area of B.

Learning B changes your belief about A.
In some cases, it does not change (events are independent), but often it does.
Conditional probability is a way to refine or update your expectations based on partial information.

Dice Rolls (Independent Events)
- A: first die is 3
- B: second die is 5
- \(P(A \mid B) = P(A) = \frac{1}{6}\) → B tells you nothing about A.
Dice and Sum (Dependent Events)
- A: first die is 3 → \(P(A) = 6/36 = 1/6\)
- C: sum of both dice is 6 → \(P(C) = 5/36\)
- Knowing the total is 6 does affect the chance the first die is 3. You have fewer valid (die1, die2) pairs to consider.
- Possible outcomes where sum is 6:
  - (1,5), (2,4), (3,3), (4,2), (5,1) → 5 outcomes
  - Only (3,3) has first die = 3 → 1 outcome
  - \(P(A \mid C) = \frac{1}{5}\)
Cards (Partial Information Impact)
- A: card is a spade (4 suits): \(P(A) = \frac{1}{4}\)
- B: card is black (2 reds, 2 blacks): \(P(B) = \frac{2}{4}= \frac{1}{2}\)
- Since spades are black, \(A \subseteq B\), so knowing the card is black narrows it to spade or club.
  
  \(P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}\)
- If card is red → \(P(\text{spade} \mid \text{red}) = 0\)

A: test result is positive
B: person has cancer
\(P(A \mid B) = 90\%\) – test catches 90% of real cases
But what we really want: \(P(B \mid A)\): the chance someone has cancer given a positive test → this is not 90% due to false positives.
This is an inverse problem: estimating a hidden cause (B) from an observed effect (A).
Solving these inverse problems requires Bayes Theorem.

It enables inference: making judgments from partial information.
Crucial for machine learning, diagnostic tests, recommendation systems, and risk analysis.
Foundation for Bayesian reasoning, which we use to update beliefs under uncertainty.