Probability is the language of uncertainty.
About this calculator
Probability is the language of uncertainty. From the chance of rain on a wedding day to the odds in a card game, the risk of system failure, or a school assignment on dice and coins — the same handful of rules apply. This calculator handles the most common combinations: joint probability (both events happen), union (either event), conditional probability (one given the other), complement (event does not happen) and the powerful at-least-once formula across many independent trials.
The 'at least once' calculation is the most surprising for beginners. If a single attempt has only a 10% chance of success, it feels like 10 attempts should give 100%. But the actual answer is 1 − (1 − 0.10)^10 = 65.1% -- not a guaranteed success, just a comfortable majority. This formula explains everything from the gambler's fallacy to system reliability engineering. It tells you why 'try enough times and it must happen' is true in the limit, but how slowly probability accumulates in real life.
The calculator distinguishes three relationships between events. Independent (default) -- the most common assumption when events are unrelated, like two coin flips. Mutually exclusive -- events that cannot co-occur, like rolling a 1 and rolling a 2 on a single die. Subset -- one event always implies the other, like 'rolled a 1' (B) implying 'rolled an odd number' (A). For more advanced combinatorial counting use the [[permutation-combination]] calculator. For algebraic equations the [[quadratic]] calculator is useful, and many probability distributions (binomial, normal) extend the geometry intuitions in the [[cone]], [[cylinder]] and [[sphere]] calculators.
Common uses
- Class 10/11/12 statistics homework -- coin tosses, dice rolls, card draws
- Estimating risk of system failure across multiple components
- Insurance and warranty pricing -- probability of at least one claim
- Marketing campaigns -- probability that a customer responds within X attempts
- Game design and lottery -- expected occurrences across many trials
Frequently asked questions
What does 'independent' actually mean?
Two events are independent when knowing one happened tells you nothing about whether the other did. Two coin flips are independent. Drawing two cards from the same deck without replacement is not independent because the first draw changes the composition of the deck.
Why is P(A and B) zero for mutually exclusive events?
By definition, mutually exclusive means they cannot both happen at the same trial. Examples: rolling a 1 vs rolling a 6 in a single die throw, or being male vs female. If they could co-occur, they wouldn
What is conditional probability and why does it matter?
P(B given A) is the probability of B happening when we already know A happened. Conditional probability is the foundation of Bayesian reasoning, medical test interpretation, spam filtering and most modern AI. It captures how new information updates our beliefs.
Why doesn't 10 attempts at a 10% chance guarantee success?
Because each attempt is independent -- the past doesn
What's the difference between odds and probability?
Probability is success / total. Odds are success / failure. A 25% probability is the same as 1:3 odds in favour. Probability ranges 0 to 1; odds range 0 to infinity. Bookmakers and betting markets use odds; statisticians and scientists use probability.