Probability calculator
This probability calculator works out the chance of an outcome: from favourable and possible cases, for two independent events, for repeated trials or when drawing without replacement. The chance of a six on a die is 1/6 (16.67%), of six correct numbers in a 6-from-49 lottery about 1 in 13.98 million.
Example: What do you want to calculate: Favourable over possible outcomes · Favourable outcomes 1 · Possible outcomes 6 → Probability: Probability: 1/6 = 16.67%. Source: Wikipedia – Probability (classical definition, independent events). Updated: .
How it is calculated
Calculating probability – the four cases
1. Classical: when all outcomes are equally likely, P = favourable outcomes ÷ possible outcomes. On a die the chance of a six is 1 ÷ 6 = 16.67%, of an even number 3 ÷ 6 = 50%.
2. Two independent events: both happen with P(A) × P(B), at least one with P(A) + P(B) − P(A) × P(B). For 50% and 30% that is 15% and 65%.
3. Repeated trials (binomial): for n independent trials with the same success probability p, P(X = k) = C(n, k) · pk · (1 − p)n−k. Exactly five heads in ten coin tosses: 24.61%. “At least one success” is 1 − (1 − p)n: at least one six in three rolls is 42.13%.
4. Drawing without replacement: each draw changes the odds, so the hypergeometric distribution applies: P = C(K, k) · C(N − K, n − k) ÷ C(N, n). In a 6-from-49 lottery three correct numbers have a 1.765% chance, six correct 1 in 13,983,816.
Tree diagrams and path rules
Along a path, probabilities multiply; different paths to the same outcome add. Cases 2 to 4 are exactly such tree diagrams, already worked out.
Honest limits
The calculator assumes independent trials with a constant probability. Dependent events (conditional probability) and continuous distributions are not included. A calculated chance says nothing about the next draw: in a lottery or with dice, chance has no memory.
Frequently asked questions
What is the formula for probability?
For equally likely outcomes: P = favourable outcomes ÷ possible outcomes. On a die the chance of a six is 1 ÷ 6.
How do I calculate probability without replacement?
With the hypergeometric distribution: P = C(K, k) · C(N − K, n − k) ÷ C(N, n). Choose “Drawing without replacement”.
How do I work with a probability tree?
Multiply the probabilities along a path and add the results of several paths that lead to the same outcome.
What are the odds of winning a 6-from-49 lottery?
About 1 in 13,983,816 for six correct numbers, roughly 0.00000715%.
How do I calculate “at least once”?
Use the complement: 1 − (1 − p)^n. The chance of at least one six in three rolls is 1 − (5/6)³ = 42.13%.
Sources and legal basis
- Wikipedia – Probability (classical definition, independent events)
- Wikipedia – Binomial distribution
- Wikipedia – Hypergeometric distribution (sampling without replacement)
As of: · Editorial policy
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