Free Probability Calculator with Worked Steps

The four rules this calculator uses

Every result on this page comes from one of four rules, and the calculator prints which rule it used next to each answer.

The multiplication rule gives the probability that both events happen. For independent events it is P(A and B) = P(A) × P(B). If the events are mutually exclusive, they cannot both happen, so P(A and B) = 0 — no multiplication needed.

The addition rule gives the probability that at least one event happens: P(A or B) = P(A) + P(B) − P(A and B). The subtraction removes the overlap so outcomes where both events occur are not counted twice. For mutually exclusive events the overlap is 0, so the rule collapses to P(A) + P(B).

The conditional probability rule gives the probability of A once you know B happened: P(A | B) = P(A and B) ÷ P(B). It rescales the joint probability by the probability of the thing you are conditioning on — which is also why it breaks down when P(B) = 0.

The complement rule is the simplest: P(not A) = 1 − P(A). Something either happens or it doesn't, and those two probabilities must add to 1.

Independent or mutually exclusive? Decide before you calculate

This is the one decision the calculator will not make for you, because it changes the formulas. Independent means one event happening tells you nothing about the other — two separate coin flips, or drawing a card and rolling a die. Mutually exclusive means the events can never happen together — a single die roll cannot land on both 2 and 5.

Students mix these up constantly, and it is worth pausing on why they are almost opposites. If A and B are mutually exclusive and B happens, you now know A did not happen — that is very strong information, the opposite of independence. Two events with nonzero probabilities can never be both independent and mutually exclusive.

A quick test for your homework problem: ask "if I knew B happened, would my estimate of A change?" If no, they are independent. Then ask "can both happen in the same trial?" If no, they are mutually exclusive. If the answer to both questions is "it depends" or "sort of," the events are dependent but overlapping, and you need P(A and B) from the problem itself — the two options on this page will not model it.

A worked example, done by hand

Suppose 30% of students in a class walk to school and 40% pack their own lunch, and the two habits are independent. So P(A) = 0.3 and P(B) = 0.4 — the calculator loads this exact example.

Both: P(A and B) = P(A) × P(B) = 0.3 × 0.4 = 0.12. Multiplying is legal only because we said the events are independent.

At least one: P(A or B) = P(A) + P(B) − P(A and B) = 0.3 + 0.4 − 0.12 = 0.58. If you skipped the subtraction you would get 0.7, which double-counts the 12% of students who do both.

Given B: P(A | B) = P(A and B) ÷ P(B) = 0.12 ÷ 0.4 = 0.3. Notice this equals P(A). That is not a coincidence — it is the definition of independence: learning that a student packs lunch tells you nothing new about whether they walk. Likewise P(B | A) = 0.12 ÷ 0.3 = 0.4 = P(B).

Neither direction of walking: P(not A) = 1 − 0.3 = 0.7. Now change the setting: if instead A and B were mutually exclusive (say, "walks to school" and "takes the bus"), then P(A and B) = 0, P(A or B) = 0.3 + 0.4 = 0.7, and P(A | B) = 0 ÷ 0.4 = 0 — once you know the student took the bus, walking is off the table.

Common mistakes to avoid

• Adding P(A) + P(B) for "or" without subtracting the overlap. That only works when the events are mutually exclusive; otherwise you count the both-happen outcomes twice.
• Multiplying P(A) × P(B) for "and" when the events are dependent. Drawing two cards without replacement is the classic trap — the second draw's probability changed, so plain multiplication is wrong.
• Treating "mutually exclusive" and "independent" as synonyms. They are nearly opposites: mutually exclusive events are maximally informative about each other.
• Swapping P(A | B) and P(B | A). "Probability of a positive test given disease" and "probability of disease given a positive test" are different numbers, often wildly different.
• Reporting a probability above 1 or below 0. If any intermediate step leaves the 0-to-1 range, an earlier assumption is broken — for mutually exclusive events, P(A) + P(B) itself can never exceed 1.
• Writing 30 when you mean 30%. A probability of 30 is meaningless; this calculator rejects it instead of guessing, so enter 0.3 or 30%.

How to check your answer

Before you box an answer on an exam, run these sanity checks — they catch most arithmetic and setup errors in a few seconds.

Every probability you computed must be between 0 and 1. P(A or B) can never be smaller than the larger of P(A) and P(B), and P(A and B) can never be larger than the smaller of them — "both" is always at least as hard as either alone. P(A) and P(not A) must add to exactly 1.

Two cross-checks tie the rules together. First, P(A and B) + P(A or B) should equal P(A) + P(B) — that is the addition rule rearranged. Second, if you claimed independence, verify P(A | B) came out equal to P(A); if it didn't, either the arithmetic slipped or the events were not independent after all.