Compute union, intersection, complement and conditional probabilities of events.
Probability quantifies how likely an event is on a scale from 0 (impossible) to 1 (certain). The addition rule gives P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive they cannot both occur, so P(A ∩ B) = 0. If they are independent, knowing B gives no information about A, so P(A ∩ B) = P(A) × P(B). Conditional probability P(A | B) = P(A ∩ B)/P(B) updates the likelihood of A given that B is known to have occurred.
Addition rule
P(A or B) = P(A) + P(B) − P(A and B)
Independent events
P(A and B) = P(A) × P(B) when A and B are independent
Conditional probability
P(A given B) = P(A and B) / P(B)
Complement
P(not A) = 1 − P(A)
Mutually exclusive events cannot happen simultaneously — if A occurs, B cannot (P(A ∩ B) = 0). Independent events can both occur, but knowing one happened gives no information about the other (P(A | B) = P(A)). These are opposite extremes: two events with P(A) > 0 and P(B) > 0 cannot be both mutually exclusive and independent.
Only when A and B are mutually exclusive, so P(A ∩ B) = 0. For any other pair of events you must subtract P(A ∩ B) to avoid double-counting outcomes where both events occur.
Yes. If A and B are positively correlated, learning that B occurred increases the probability of A. For example, the probability of a positive cancer test is higher given that the patient actually has cancer than in the general population.