Probability Calculator

① Complement: P(not A)

The probability that event A does not happen.

P(not A) = 1 − P(A)

Example: if P(rain) = 0.3, then P(no rain) = 1 − 0.3 = 0.7

② Intersection: P(A and B) — independent events

Both A and B occur. Works when the events do not influence each other.

P(A ∩ B) = P(A) × P(B)

Example: P(heads) = 0.5, P(six) = 1/6 ⇒ P(heads and six) = 0.5 × 0.167 ≈ 0.083

③ Union: P(A or B) — mutually exclusive

Either A or B occurs; they cannot both happen at the same time.

P(A ∪ B) = P(A) + P(B)

Example: rolling a 1 or a 2 on a die: 1/6 + 1/6 = 2/6 ≈ 0.333

④ Union: P(A or B) — not mutually exclusive

Either A or B occurs; the events can overlap, so we subtract the overlap.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

For independent events P(A ∩ B) = P(A) × P(B).

Example: P(A)=0.4, P(B)=0.5 ⇒ P(A or B) = 0.4 + 0.5 − 0.2 = 0.7

⑤ Conditional: P(A|B)

The probability that A occurs given that B has already occurred.

P(A|B) = P(A ∩ B) / P(B)

Example: P(A ∩ B)=0.12, P(B)=0.4 ⇒ P(A|B) = 0.12 / 0.4 = 0.3

Sample Space & Events

The sample space S is the set of all possible outcomes. An event A is a subset of S. P(S) = 1.

Law of Total Probability

If B₁, B₂, … partition S, then P(A) = ∑ P(A|B_i) × P(B_i).

Bayes' Theorem

P(A|B) = P(B|A) × P(A) / P(B) — updates the probability of A given new evidence B.