When do I add vs multiply probabilities?

Add for OR (subtract the overlap unless mutually exclusive); multiply for AND when events are independent.

Mutually exclusive vs independent?

Exclusive = can’t both happen (overlap 0). Independent = one doesn’t affect the other. They’re different ideas.

Works out the chance of A or B, A and B, and if they affect each other.

P(A∪B) = P(A) + P(B) − P(A∩B)

Probability is the chance of something, a number from 0 (never) to 1 (always); the intersection A and B is the overlap where both events happen at once. (e.g. a card that is a king AND a heart)
The union A or B is everything in either circle — A, B, or both together. (e.g. a card that is a king OR a heart)
Mutually exclusive events cannot both happen, so their overlap is empty: P(A and B) = 0. (e.g. rolling a 2 and a 5 at once)
When events are mutually exclusive the OR rule simplifies to just adding them. (e.g. P(A or B) = 0.3 + 0.4 = 0.7)
A set of events is exhaustive when together they cover every possible outcome — their union is everything that could happen. (e.g. heads or tails on a coin)
For exhaustive events the probabilities add up to 1; if they are also mutually exclusive they form a partition (a clean split). (e.g. P(A) + P(B) + ... = 1)
The complement rule says P(not A) — the chance A does not happen — is 1 minus P(A). (e.g. 1 - 0.3 = 0.7)
For independent events (one does not affect the other), multiply to get the AND probability. (e.g. 0.5 x 0.5 = 0.25)
The general OR rule subtracts the intersection: P(A or B) = P(A) + P(B) - P(A and B). (e.g. 0.5 + 0.4 - 0.2 = 0.7)
Two coins both landing heads shows the multiply rule in action. (e.g. 0.25)
Every probability stays between 0 and 1 inclusive. (e.g. 0 <= P <= 1)

$P (A \cup B) = P (A) + P (B) - P (A \cap B)$

Some kids like pizza, some like ice cream, some like both. The two circles show who likes what, and where they overlap is the "both" group.

In a class, 50% like pizza, 40% like ice cream, and 20% like both. probrules gives the chance of liking either as 0.5 + 0.4 − 0.2 = 70%, subtracting the overlap so it is not double-counted.
Draw one card: P(king) = 4/52, P(heart) = 13/52, and P(king of hearts) = 1/52. The chance of a king or a heart is 4/52 + 13/52 − 1/52 = 16/52 ≈ 31%.
If P(pet) = 0.5, P(sport) = 0.4, and P(both) = 0.2, then since 0.5 × 0.4 = 0.2 matches the overlap, probrules confirms the two are independent — knowing one tells you nothing about the other.

If P(A) = 0.5, P(B) = 0.4, and P(A∩B) = 0.2, then A or B happens 70% of the time, and the two are independent.

When do I add vs multiply probabilities?: Add for OR (subtract the overlap unless mutually exclusive); multiply for AND when events are independent.
Mutually exclusive vs independent?: Exclusive = can’t both happen (overlap 0). Independent = one doesn’t affect the other. They’re different ideas.