Independence is an assumption, not a given
Multiplying probabilities to get "both" only works when the events are independent — when one occurring tells you nothing about the other. Rain today and rain tomorrow are not independent, and treating them as such badly understates the joint probability.
The "either or" calculation subtracts the overlap (A × B) because otherwise the cases where both happen would be counted twice.
Repetition turns small chances into large ones
A 5% chance sounds negligible until it is repeated. Across twenty independent trials, the chance of it happening at least once rises to about 64%. This is the arithmetic behind why rare failures become routine at scale.
The formula is 1 minus the probability of it never happening: 1 − (1 − p)ⁿ.
Conditional probability
When events are dependent, the probability of both is P(A) times P(B given A) — not P(A) times P(B). Drawing two aces from a deck without replacement is 4/52 times 3/51, because the first draw changes what remains.
Confusing conditional and unconditional probability is behind many statistical errors, including the prosecutor's fallacy: the chance of the evidence given innocence is not the chance of innocence given the evidence.
The gambler's fallacy
Independent events have no memory. A coin that has landed heads ten times running still has a 50% chance of heads on the eleventh toss. The coin does not owe anyone a tail.
What is true is that long runs are unlikely in advance. The probability of ten heads before you start is about one in a thousand — but once nine have landed, the tenth is still an even bet.
Frequently asked questions
How do odds relate to probability? +
Odds of 3:1 in favour mean three successes for every one failure, which is a probability of 3/4 or 75%. Probability is successes over total; odds are successes over failures.
What does independence mean? +
That one event happening tells you nothing about whether the other will. Two dice are independent; drawing cards without replacement is not.
Why do small probabilities feel impossible? +
Because we evaluate them singly. A one-in-a-million event happening to someone in a city of ten million is expected roughly ten times over, which is why coincidences are far more common than intuition suggests.