The definition of probability is simple. P(A) = (number of desired outcomes) ÷ (total number of outcomes). If a bag has 12 balls and 3 of them are red, the probability of drawing a red ball is 3/12. This much isn't hard. The problem is that as soon as things get even a little more complicated — like "the probability that at least one is ~" — applying this definition directly suddenly becomes difficult.
For example, let's find "the probability that at least one 6 shows up when you roll 3 dice." Counting it directly means finding "exactly 1 die shows a 6" + "exactly 2 dice show a 6" + "all 3 dice show a 6," then adding them together. The more cases there are, the messier this calculation gets.
This is exactly where a complementary event comes in. The event that's the exact opposite of some event A (A not happening) is called its "complementary event," and since the two together always make up everything (probability 1), P(A) = 1 − P(complement of A) holds. The complement of "at least one 6" is "none of them are 6." Much easier to calculate — the probability a single die isn't a 6 is 5/6, and for all 3 dice to not be 6, you just multiply 5/6 by itself three times: (5/6)³ = 125/216. Subtract that from 1, and 91/216 is exactly the answer to "at least one 6."
It's a great habit to suspect a complementary event whenever you see a phrase like "at least," "at minimum once," or "one or more" in a problem. The more a situation splits into many messy cases when you count directly, the more often the opposite situation ("none at all," "all failed") wraps up with a single simple multiplication. This isn't luck — it comes from a basic property of probability: an event and its complement together always make up the entire set of outcomes (probability 1).
On our activity page, you can first check the basic definition of probability with a simple example of drawing balls, then increase the number of dice with a slider and watch the probability of "at least one 6" come out instantly through a complementary-event calculation — and watch that probability get closer and closer to 1 as the number of dice increases.