📘 Basics of Probability

Sample space (S):

The set of all possible outcomes.

Event:

A subset of S (i.e., A⊆SA \subseteq SA⊆S).

Probability measure P:

A function assigning values to events, satisfying:

• 0≤P(A)≤10 \leq P(A) \leq 10≤P(A)≤1

• P(S)=1P(S) = 1P(S)=1

• For disjoint events A1,A2,…:

  P(A1∪A2∪… )=P(A1)+P(A2)+

🔄 Conditional Probability

Definition:

Given P(B)>0P(B) > 0P(B)>0, the conditional probability of A given B is:

P(A | B) = P(A ∩ B) / P(B)

Key Properties:

• P(A | B) updates belief in A after observing B.

• If A and B are independent:

  P(A | B) = P(A)
  P(A ∩ B) = P(A) * P(B)