Probability Calculator

What should I know about 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.

What do I need to know about 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.

What do I need to know about 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.

What's the key thing to understand about 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.

Worked Example: Drawing Two Cards Without Replacement

From a standard 52-card deck, what's the probability of drawing two hearts in a row without replacing the first card? There are 13 hearts in the deck. P(1st card is a heart) = 13/52 P(2nd card is a heart | 1st was a heart) = 12/51 (one heart and one card total removed) By the multiplication rule for dependent events: P(both hearts) = (13/52) × (12/51) = 156/2652 ≈ 0.0588, or about 5.88%. Compare this to drawing WITH replacement (putting the card back and reshuffling before the second draw), where the two draws are independent: P(both hearts) = (13/52) × (13/52) = 169/2704 ≈ 0.0625, or 6.25% — slightly higher, because without replacement, removing a heart on the first draw makes the deck slightly 'heart-poorer' for the second draw. This distinction between independent and dependent events is the single most common source of error in probability problems, and the phrase 'without replacement' is the signal to update the probability for the second event based on what happened in the first.

Probability Rules Quick Reference

A fast way to check which rule applies: ask whether the word in the question is 'and' (usually multiplication) or 'or' (usually addition), then ask whether the events could happen at the same time (for 'or' problems) or whether one affects the other (for 'and' problems). Getting this classification right is most of the work in any probability question — the arithmetic afterward is usually simple once the right rule is chosen.

Worked Example: The Addition Rule With Overlap

What is the probability of drawing a card that is either a heart or a king from a standard 52-card deck? These events are NOT mutually exclusive, because the king of hearts is both. P(heart) = 13/52, P(king) = 4/52, P(heart and king) = 1/52 (just the king of hearts) P(heart or king) = P(heart) + P(king) - P(heart and king) = 13/52 + 4/52 - 1/52 = 16/52 ≈ 0.3077, or about 30.8%. Without subtracting the overlap, you'd get 17/52 ≈ 32.7% — double-counting the king of hearts, which was included in both the 'heart' count and the 'king' count. This subtraction of the overlap (the intersection) is the entire difference between the two versions of the addition rule, and forgetting it is one of the most common probability mistakes made even by otherwise careful students.