Probability Calculator

Probability is a number between 0 and 1 (or equivalently, 0% and 100%) that describes how likely an event is to happen. A probability of 0 means the event is impossible; a probability of 1 (100%) means it is certain. Everything in between describes varying degrees of likelihood — a probability of 0.5 (50%) means the event is exactly as likely to happen as not, like a fair coin landing on heads. The most basic probability calculation is for a single event with equally likely outcomes: Probability = Favorable outcomes ÷ Total outcomes. Rolling a standard six-sided die and wanting a 4 has 1 favorable outcome out of 6 total possible outcomes, so the probability is 1/6, or about 16.7%. Drawing a red card from a standard 52-card deck has 26 favorable outcomes (the red cards) out of 52 total, giving a probability of 26/52 = 0.5, or 50%. Things get more interesting when you combine two events. The word "AND" in probability means both events need to happen, while "OR" means at least one of them needs to happen — and the formulas for combining probabilities differ significantly between the two. If two events are independent, meaning the outcome of one has no effect on the outcome of the other, the probability of both happening (A AND B) is simply the product of their individual probabilities: P(A and B) = P(A) × P(B). For example, if the probability of rain tomorrow is 30% and, independently, the probability your favourite team wins is 60%, the probability of both happening is 0.30 × 0.60 = 0.18, or 18%. The OR case requires a bit more care because of double-counting. If you simply added P(A) + P(B), you would count the situation where both A and B happen twice — once within P(A) and once within P(B). The correct formula subtracts that overlap back out: P(A or B) = P(A) + P(B) − P(A and B). Using the same independent-events assumption, this becomes P(A or B) = P(A) + P(B) − P(A) × P(B). With a 30% chance of rain and a 60% chance of your team winning, the probability that at least one of those things happens is 0.30 + 0.60 − 0.18 = 0.72, or 72%. The independence assumption is critical and is the single biggest misconception around these calculations. The AND and OR formulas shown above only apply cleanly when the two events genuinely do not influence each other. If the events are dependent — for example, drawing two cards from a deck without replacing the first one — these simple multiplication and addition formulas no longer apply directly, and you need conditional probability instead, which accounts for how the first event changes the odds of the second. This calculator explicitly assumes independence in its AND and OR modes and states that assumption in the hint text beneath the result, so you always know what is (and is not) being calculated. Another important edge case: probability calculations break down if the number of "favorable" outcomes somehow exceeds the total possible outcomes, which would imply a probability over 100% — a logical impossibility. This calculator catches that mistake and shows a clear validation message instead of a misleading number above 100%.