In more detail
For equally likely outcomes, probability is favourable outcomes divided by total outcomes. The complement rule is P(not A) = 1 − P(A). For independent events, P(A and B) = P(A) × P(B). For "or", add and remove the overlap: P(A or B) = P(A) + P(B) − P(A and B). Conditional probability, P(A | B) = P(A and B) / P(B), is the chance of A once you know B happened. See How Randomness and Fair Draws Work.
Why it matters
Intuition is unreliable with chance. Independent events have no memory: after five reds on a fair roulette wheel, red is no less likely on the sixth spin (the gambler's fallacy). Coincidences are far more common than they feel, because there are many ways for one to happen. And forgetting the overlap in an "or" question double-counts.
Example
at least one 6 in four rolls 1 − (5/6)⁴ = 0.5177
ace or heart from 52 cards 4/52 + 13/52 − 1/52 = 16/52 = 0.3077
two sixes with two dice 1/6 × 1/6 = 1/36 = 0.0278
shared birthday among 23 people 1 − (365 × 364 × … × 343) / 365²³ = 0.5073
It is often easiest to calculate "none" and subtract from 1. With 50 people the birthday probability rises to 0.9704.