Probability Calculator
Combine two independent probabilities to get the chance of both, either or neither happening.
Formula and sources updated · Data sources · 1View algorithm, constants and default inputs (JSON)
About this calculator
For independent events, the probability of both happening is the product of the individual probabilities, and the probability of either is the sum minus the overlap. Complement gives the chance of neither.
Worked example
calculated with the default values below
Probability of A 0.5 · Probability of B 0.3 Both A and B 0.150000 · Either A or B 0.650000 · Neither 0.350000 · Only A 0.350000 · Odds of both 0.1765
Method and assumptions
P(A∩B) = P(A)P(B); P(A∪B) = P(A) + P(B) − P(A∩B); O = P(A∩B) ÷ (1 − P(A∩B)) Assumes independent events and probabilities from 0 to 1. Odds are p ÷ (1 − p); when p = 1, odds are infinite and shown as an em dash. For independent events, the probability of both happening is the product of the individual probabilities, and the probability of either is the sum minus the overlap. Complement gives the chance of neither. The references explain definitions and methods; any extra preset must be checked against the assumptions on this page.
Data sources
The references explain definitions and methods; any extra preset must be checked against the assumptions on this page.
Disclaimer
These calculators give general mathematical results. They suit study, planning and estimation — verify anything that feeds a graded submission, a published figure or a financial commitment.
Implementation source
This record describes the implementation used by this page. It is not independent validation. External references, when available, support only the stated formula or definition; estimates and custom rules retain their own assumptions.
Formula and variables
P(A∩B) = P(A)P(B); P(A∪B) = P(A) + P(B) − P(A∩B); O = P(A∩B) ÷ (1 − P(A∩B))
- P(A)
- Probability of A
- P(B)
- Probability of B
- P(A∩B)
- Both occur
- P(A∪B)
- Either occurs
- O
- Odds for both
Assumes independent events and probabilities from 0 to 1. Odds are p ÷ (1 − p); when p = 1, odds are infinite and shown as an em dash.
Frequently asked questions
Why does this calculator assume A and B are independent events?
The formulas multiply and add probabilities, which is only valid when A and B do not influence each other. At the default 0.5 and 0.3 the chance of both is 0.15 and either is 0.65, but if drawing two cards without replacement the second depends on the first and those numbers break. The tool reports independent odds; use it for coin flips and dice, not linked events. When both probabilities are 1, the probability of both is certain and the odds are infinite. Since the calculator displays finite numeric results, it shows an em dash for that odds value.
Which inputs and limits apply to Probability Calculator?
Use the units shown beside each field. These are form limits, not recommended settings. Probability of A: 0–1; Probability of B: 0–1 P(A∩B) = P(A)P(B); P(A∪B) = P(A) + P(B) − P(A∩B); O = P(A∩B) ÷ (1 − P(A∩B)) Assumes independent events and probabilities from 0 to 1. Odds are p ÷ (1 − p); when p = 1, odds are infinite and shown as an em dash. For independent events, the probability of both happening is the product of the individual probabilities, and the probability of either is the sum minus the overlap. Complement gives the chance of neither. The references explain definitions and methods; any extra preset must be checked against the assumptions on this page.
What is a complete worked example for Probability of A = 0.5; Probability of B = 0.3?
Inputs: Probability of A = 0.5; Probability of B = 0.3. Calculation: P(A∩B) = P(A)P(B); P(A∪B) = P(A) + P(B) − P(A∩B); O = P(A∩B) ÷ (1 − P(A∩B)). Results: Both A and B = 0.150000; Either A or B = 0.650000; Neither = 0.350000; Only A = 0.350000; Odds of both = 0.1765.