At its core, this calculator takes two probability values — P(A) and P(B), each between 0 and 1 — and applies the fundamental rules of probability theory to produce three combined outcomes. The key assumption is that the two events are independent, meaning the occurrence of one has absolutely no influence on the occurrence of the other. This is a critical distinction: independence is what allows us to multiply probabilities directly, and violating this assumption is one of the most common errors in real-world probability reasoning.

For P(A and B), the calculator uses the multiplication rule for independent events: P(A and B) = P(A) × P(B). This tells you the probability that both events happen simultaneously. For example, if there is a 30% chance of rain and a 50% chance of a traffic jam, and these events are independent, then the chance of experiencing both is 0.30 × 0.50 = 0.15, or 15%. This figure is always less than or equal to both individual probabilities, which is why simultaneous rare events become extremely unlikely very quickly.

For P(A or B), the calculator applies the addition rule with the inclusion-exclusion correction: P(A or B) = P(A) + P(B) − P(A and B). Without subtracting the overlap, you would double-count scenarios where both events occur, inflating the result. This formula gives the probability that at least one of the two events takes place. P(neither A nor B) is then derived as 1 − P(A or B), representing the chance that both events fail to occur — a useful figure in risk assessment and quality control contexts where you want to know the probability of a clean, event-free outcome.

A common mistake is confusing 'mutually exclusive' events with 'independent' events. Mutually exclusive events cannot both happen (P(A and B) = 0), while independent events simply do not influence each other but can absolutely co-occur. Using independence formulas on dependent events — like drawing two cards without replacement — produces incorrect results. Always verify your independence assumption before entering values. Additionally, remember that probabilities must be expressed as decimals or true fractions between 0 and 1; entering percentages without converting them (e.g., typing 30 instead of 0.30) will yield meaningless outputs.