Probability Calculator
Single-event, 'A and B', 'A or B', at-least-one and binomial (exactly k of n) probabilities as percentages, fractions and odds, with the rule applied shown.
Probability Calculator: with the default inputs, probability is 16.6667%.
Outcomes that count as a success.
Probability of the first event, as a percent.
Used only for the binomial mode.
- As a decimal
- 0.166667
- Odds
- 1 to 5 in favor (5 to 1 against)
- Probability it does NOT happen
- 83.3333%
- P(at least k successes)
- —Binomial mode only.
- P(at most k successes)
- —Binomial mode only.
Assumptions
- The 'and' / 'or' modes assume A and B are independent.
- Repeated tries are independent with the same success probability each time (Bernoulli trials).
- Only the inputs relevant to the selected mode are used.
How this is worked out
The formula
P(A) = favorable outcomes ÷ total equally likely outcomes Independent events: P(A and B) = P(A) × P(B) P(A or B) = P(A) + P(B) − P(A and B) P(at least one in n tries) = 1 − (1 − p)ⁿ Binomial: P(exactly k of n) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ Odds in favor = P ÷ (1 − P)
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- What do you want to find?
- Choose one of 5 options.P(A) — favorable outcomes ÷ total outcomes · P(A and B) — both happen (independent) · P(A or B) — at least one happens (independent) · At least one success in n tries · Exactly k successes in n tries (binomial)
- Favorable outcomes
- Outcomes that count as a success.0 or more · whole numbers only · defaults to 1
- Total equally likely outcomes
- A number.1 or more · whole numbers only · defaults to 6
- P(A)
- Probability of the first event, as a percent.a percentage · from 0 to 100 · defaults to 50
- P(B)
- A number.a percentage · from 0 to 100 · defaults to 50
- Probability of success per try
- A number.a percentage · from 0 to 100 · defaults to 16.6667
- Number of tries (n)
- A number.from 1 to 1000 · whole numbers only · defaults to 10
- Successes wanted (k)
- Used only for the binomial mode.from 0 to 1000 · whole numbers only · defaults to 3
What you get back
- Probabilitymain answer
- As a decimal
- Odds
- Probability it does NOT happen
- P(at least k successes)
- Binomial mode only.
- P(at most k successes)
- Binomial mode only.
What this assumes
- The 'and' / 'or' modes assume A and B are independent.
- Repeated tries are independent with the same success probability each time (Bernoulli trials).
- Only the inputs relevant to the selected mode are used.
About this calculator
Probability measures how likely something is on a scale from 0 (impossible) to 1 (certain), usually quoted as a percentage. Almost every probability problem you'll meet is one of five patterns, and this calculator handles each one and shows which rule it used.
The five patterns
- One event. If every outcome is equally likely, P = favorable ÷ total. A die has 6 faces, one is a 4, so P(4) = 1/6 ≈ 16.7%.
- Both A and B (independent events, meaning one doesn't affect the other): multiply. Two heads in a row: 0.5 × 0.5 = 25%.
- A or B: add, then subtract the overlap, because outcomes where both happen were counted twice. Heads on either of two flips: 0.5 + 0.5 − 0.25 = 75%.
- At least one success in n tries: the complement trick. It's far easier to find the chance of zero successes, (1 − p)ⁿ, and subtract from 1. At least one six in 4 rolls: 1 − (5/6)⁴ ≈ 51.8%.
- Exactly k successes in n tries (binomial): count the arrangements with C(n, k), multiply by the chance of any one arrangement, pᵏ(1 − p)ⁿ⁻ᵏ. The calculator also gives "at least k" and "at most k" and plots the whole distribution.
Probability vs. odds
Probability compares successes to all outcomes; odds compare successes to failures. A 1/6 probability is odds of 1 to 5 in favor (or 5 to 1 against). Betting and medicine both quote odds, so the calculator gives both.
Independence matters
The multiplication rule is only for independent events. Drawing two aces from a deck without replacement is 4/52 × 3/51, not (4/52)² — the second draw depends on the first. When events are dependent you need conditional probabilities, which is a different calculation.
Common mistakes
- Adding probabilities for "or" without subtracting the overlap (it can push you past 100%).
- Assuming a 1-in-10 chance means 10 tries guarantee success. It's 1 − 0.9¹⁰ ≈ 65%.
- Mixing up "at least k" with "exactly k". Check the extra outputs in binomial mode.
Frequently asked questions
▸How do I calculate the probability of A and B?
If the events are independent, multiply: P(A and B) = P(A) × P(B). Two coin flips both heads: 0.5 × 0.5 = 0.25 = 25%.
▸How do I calculate the probability of A or B?
Add the probabilities and subtract the chance both happen: P(A) + P(B) − P(A and B). For independent events P(A and B) = P(A)P(B). If the events can't both happen, the overlap is zero and you just add.
▸What is the 'at least one' formula?
1 − (1 − p)ⁿ, where p is the chance of success on each try and n is the number of tries. It works by finding the chance of no successes at all and subtracting from 1.
▸How do I convert probability to odds?
Odds in favor = P ÷ (1 − P). A probability of 0.2 is odds of 0.25 to 1, usually written 1 to 4 in favor or 4 to 1 against.
▸What's the binomial probability formula?
P(exactly k of n) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ. C(n, k) counts which k of the n tries succeed; the rest is the probability of one such pattern.
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