What is probability?
A number between 0 and 1 that measures how likely an event is — and the rules that govern how those numbers combine.
The three axioms
Probability is built on three rules. For any probability P:
1. P(A) ≥ 0 for every event A. 2. P(Ω) = 1, where Ω is the whole sample space. 3. If A and B are disjoint, P(A ∪ B) = P(A) + P(B).Everything else — conditional probability, independence, Bayes' rule — is a consequence of these three. The axioms are what turn "probability" from a vague feeling into a precise tool.
Conditional probability
The probability of A given that B has happened:
P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.Conditional probability is how you update: knowing something (B happened) changes what you should believe about something else (A).
Worked example
A fair die is rolled twice. What is the probability that both rolls are 4 or greater, given that the first roll was at least 4?
Sample space (given first ≥ 4): {4, 5, 6} for the second roll. Favorable outcomes: both rolls in {4, 5, 6} → 3 × 3 = 9 outcomes. Total outcomes with first ≥ 4: 3 × 6 = 18. P(both ≥ 4 | first ≥ 4) = 9 / 18 = 0.5.Knowing the first roll was ≥ 4 cuts the sample space in half — and the second roll is then exactly a 50/50 question.
Flashcard · Unit 2
Update your belief about A given B.
Check your understanding
A bag has 3 red and 2 blue marbles. You draw one without looking. What is the probability it is red?