Probability Calculator

P(A AND B) — Independent Events

The probability that two events both happen depends on whether they're independent. For independent events — where the outcome of one has no effect on the other — you multiply their probabilities: P(A and B) = P(A) × P(B). A clear example: rolling a 6 on a die twice in a row is 1/6 × 1/6 = 1/36, because the second roll is unaffected by the first. Tossing two heads is 1/2 × 1/2 = 1/4. The multiplication rule extends to more events: the chance of three independent things all happening is the product of all three probabilities. The crucial condition is independence — this multiplication only works when the events genuinely don't influence each other. Drawing two aces from a deck without replacing the first is NOT independent: after drawing one ace, only 3 aces remain among 51 cards, so the second probability changes (this is dependent, and you'd use conditional probability). A common error is multiplying probabilities for events that are actually linked. Independence also explains why the chance of a long run of the same outcome gets small fast (five heads in a row is 1/32), even though each individual toss remains 50/50 — the 'gambler's fallacy' is wrongly believing past independent results change future ones. This calculator applies the correct rule based on whether events are independent.

P(A OR B) — Addition Rule

The probability that at least one of two events happens uses the addition rule: P(A or B) = P(A) + P(B) − P(A and B). You add the individual probabilities but then subtract the overlap (the probability of both happening), because otherwise the overlap gets counted twice. A worked example: the probability of drawing a card that's either a heart or a face card from a standard deck — there are 13 hearts and 12 face cards, but 3 cards are both (the heart face cards), so P = 13/52 + 12/52 − 3/52 = 22/52. For mutually exclusive events — ones that cannot both happen at once — the overlap is zero, so the rule simplifies to P(A or B) = P(A) + P(B). For example, rolling a die and getting either a 2 or a 5 is 1/6 + 1/6 = 2/6, since a single roll can't be both. The most common mistake is forgetting to subtract the intersection for events that can overlap, which inflates the probability. Another check: a probability can never exceed 1 (certainty), so if your addition gives more than 1, you've likely failed to subtract an overlap or misidentified the events. This calculator handles both the general case and the mutually-exclusive case.

Conditional Probability

Conditional probability is the chance of one event given that another has already occurred, written P(A given B) and calculated as P(A and B) ÷ P(B). It captures how knowing one piece of information updates a probability. The classic real-world lesson involves medical testing: even a very accurate test can give a surprisingly high rate of false positives when a condition is rare in the population, because the few true positives are swamped by false positives from the large healthy majority. This is why a positive result on a screening test for a rare condition often warrants confirmatory testing rather than immediate alarm — the probability of actually having the condition given a positive test (which is what you care about) can be much lower than the test's headline accuracy suggests. This counterintuitive result comes from Bayes' theorem, which formalises how to update probabilities with new evidence. Conditional probability matters in many fields: weather forecasting, spam filtering, insurance, and risk assessment all rely on updating probabilities as information arrives. A key insight is that P(A given B) and P(B given A) are generally different — confusing the two (the 'prosecutor's fallacy') causes serious reasoning errors, including in legal cases. The takeaway is that probabilities should be revised in light of relevant conditions, and that base rates (how common something is to begin with) matter enormously.

Exam Tips and Common Errors

A few principles prevent the most common probability mistakes. First, always check whether events are independent before multiplying — if one event affects the other (like drawing cards without replacement), you must use conditional probability, not simple multiplication. Second, for 'or' problems, remember to subtract the overlap unless the events are mutually exclusive; forgetting this double-counts and can give a probability over 1, which is impossible and a useful error-check. Third, distinguish 'with replacement' from 'without replacement' in drawing problems, as it changes whether probabilities stay constant. Fourth, the complement rule is often the easy route: P(at least one) = 1 − P(none), which is far simpler than adding many cases — for example, the chance of at least one head in several tosses is easiest as 1 minus the chance of all tails. Fifth, every probability lies between 0 (impossible) and 1 (certain), so any answer outside that range signals an error. Keep fractions, decimals, and percentages consistent and convert carefully (1/4 = 0.25 = 25%). Watch order of operations, and in tree diagrams, multiply along branches and add across them. Finally, sanity-check against intuition where possible — though, as conditional probability shows, intuition can mislead, so trust the correct rule over a gut feeling when they conflict.