Calculate the probability of two independent events happening together, either occurring, or not occurring.
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Probability measures how likely a given event is to occur, and always ranges between 0 and 1. When dealing with two independent events A and B, the most important derived probabilities are calculated as follows: the probability of both events occurring together, P(A∩B), equals the product of their individual probabilities; the probability of at least one occurring, P(A∪B), equals the sum of their probabilities minus the probability of both occurring together; and the probability of the event not occurring, P(not A), equals 1 minus its probability. These rules are widely used in games, drawings, and risk analysis, and form the theoretical basis for all more complex probability calculations. The independence assumption behind the P(A∩B) formula is worth checking carefully in real situations, since two events that influence each other — drawing two cards from a deck without replacement, for instance — require a different, conditional approach rather than simply multiplying their individual probabilities.
Probability theory can grow mathematically sophisticated very quickly, but the three foundational rules covered here — for combined events, either-or events, and complementary events — are enough to reason correctly about a surprisingly large share of everyday probability questions, from card games to basic risk assessment.
The multiplication rule, P(A∩B) = P(A) × P(B), applies to independent events: two events where the outcome of one has no effect on the probability of the other. Rolling a die and flipping a coin are independent — knowing the die landed on 4 tells you nothing about how the coin will land. This is why the probability of rolling a 6 and flipping heads together is simply 1/6 multiplied by 1/2, giving 1/12.
The critical caveat is that this multiplication rule only holds for genuinely independent events. Drawing two cards from a deck without putting the first one back changes the composition of the remaining deck, making the two draws dependent — the correct calculation requires conditional probability (adjusting the second event's probability based on the first event's outcome), not simple multiplication of the original, unconditional probabilities.
The addition rule, P(A∪B) = P(A) + P(B) − P(A∩B), calculates the probability that at least one of two events happens. The subtraction of the overlap term matters because simply adding P(A) and P(B) would double-count any outcome where both events occur simultaneously — rolling either a 4 or an even number on a die would double-count the number 4 itself if the overlap weren't subtracted back out.
The complement rule, P(not A) = 1 − P(A), is often the fastest route to an answer for 'at least one' problems involving multiple trials, where directly calculating every winning scenario would be tedious. The probability of rolling at least one six in four dice rolls is far easier to find by calculating the probability of rolling zero sixes across all four rolls, then subtracting that result from 1, than by separately calculating the probability of exactly one, two, three, and four sixes and adding them together.
These three rules together underpin games of chance, insurance risk pricing, quality control sampling, and genetics (where the probability of inheriting particular combinations of traits follows the same independence and combination logic), making them some of the most broadly applicable formulas in all of applied mathematics.
It means the outcome of one event has no effect on the probability of the other, such as two separate coin flips.
No, valid probabilities are always between 0 and 1 (or 0% and 100%).
For dependent events, P(A∩B) requires the conditional probability P(B|A) instead of simply multiplying P(A) and P(B).