1. The Concept Before the Formula: Updating Probability with Partial Information
Core Idea: When you learn that some event has occurred, you throw away every outcome that contradicts that information. The sample space shrinks to the known event, and you recount the favorable outcomes inside that smaller space.
Before writing any formula, let's build the idea from scratch with three concrete examples.
Counted Example A — A Die Roll with Parity Disclosed
Roll a fair six-sided die. The full sample space is $S = \{1,2,3,4,5,6\}$, with each outcome equally likely (probability $\tfrac{1}{6}$).
Question: What is the probability the result is at most 3, given that the result is odd?
- Define $A = \{1,2,3\}$ (result is at most 3) and $B = \{1,3,5\}$ (result is odd).
- Once you learn the result is odd, every even outcome ($2, 4, 6$) is impossible. The sample space shrinks from $\{1,2,3,4,5,6\}$ to $\{1,3,5\}$.
- Inside this reduced space, which outcomes also belong to $A$? They are $\{1,3\}$ — two out of three equally likely outcomes.
$$P(A \mid B) = \frac{\text{favorable outcomes in reduced space}}{\text{total outcomes in reduced space}} = \frac{2}{3}$$
Without the information, $P(A) = \tfrac{3}{6} = \tfrac{1}{2}$. The partial information changed the probability from $\tfrac{1}{2}$ to $\tfrac{2}{3}$.
Counted Example B — A Card Draw with a Suit Revealed
Draw one card from a standard 52-card deck.
Question: What is the probability the card is a King, given that it is a Heart?
- Define $A = \{\text{card is a King}\}$ (4 Kings in the deck) and $B = \{\text{card is a Heart}\}$ (13 Hearts).