Probability calculator
Probability rules are easier to trust when you can see the overlap. Set two events and the Venn diagram shows every combination.
Circle areas are drawn to scale. Pick a result to shade it.
- P(A and B) = P(A) × P(B) = 0.5 × 0.4 = 0.2000
- P(A or B) = P(A) + P(B) − P(A and B) = 0.5 + 0.4 − 0.2 = 0.7000
- P(A | B) = P(A and B) / P(B) = 0.2 / 0.4 = 0.5000
The rules, and what they look like
- Complement: P(not A) = 1 − P(A). Everything outside circle A.
- Addition rule: P(A or B) = P(A) + P(B) − P(A and B). Adding the two circles counts the overlap twice, so remove it once.
- Multiplication rule: P(A and B) = P(A) × P(B | A), which simplifies to P(A) × P(B) when A and B are independent.
- Conditional probability: P(A | B) = P(A and B) / P(B). Restrict attention to circle B and ask what share of it is also in A.
Independent is not the same as mutually exclusive
Mutually exclusive events cannot happen together, so their circles do not touch. Independent events can happen together, and knowing one happened does not change the chance of the other. In fact, two mutually exclusive events with non-zero probability are always dependent: if A happened, B certainly did not.
Bayes' theorem and the base rate
With a condition affecting 1% of people, a test that catches 90% of cases and gives 9% false positives, only about 9% of people who test positive actually have it. The 1,000-person grid shows why: false positives from the large healthy group outnumber the true positives. Gerd Gigerenzer's research found that this "natural frequency" picture helps doctors get such questions right far more often than the formula alone.
Want to count outcomes directly? Try the dice probability calculator, or test your intuition on the Monty Hall problem.
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Questions students ask
How do you find the probability of A or B?
P(A or B) = P(A) + P(B) − P(A and B). You subtract the overlap because it was counted in both P(A) and P(B). If the events are mutually exclusive, the overlap is 0.
How do you find the probability of A and B?
For independent events, multiply: P(A and B) = P(A) × P(B). In general, P(A and B) = P(A) × P(B | A).
What is conditional probability?
P(A | B), the probability of A given that B happened, equals P(A and B) ÷ P(B). You shrink the world to the B circle and ask what fraction of it is also A.
What is Bayes' theorem used for?
Reversing a conditional probability: from P(positive test | disease) to P(disease | positive test). It is why a positive result for a rare condition can still leave the chance of having it well below 50%.