Probability Calculator
This calculator works out probabilities across eight common scenarios, all in one tool: basic probability, combined AND/OR events, conditional probability, Bayes' theorem, permutations and combinations, binomial distribution, normal distribution, and Poisson distribution. Select the tab that matches your question, then enter the figures it asks for: favourable and total outcomes for a simple probability, P(A) and P(B) with whether they are independent or mutually exclusive for combined events, P(A and B) and P(B) for conditional probability, a prior probability with sensitivity and false positive rate for Bayes' theorem, n and r for combinations and permutations, trials, probability and successes for binomial, mean, standard deviation and a value for normal (including z-scores), or an average rate and count for Poisson. Each tab returns the exact probability as a decimal and a percentage, plus a plain-English breakdown of the formula used. The binomial and Poisson tabs also draw a bar chart and table of the full distribution, and the normal tab plots a bell curve showing the area calculated. Every tab includes a worked example so you can check your own numbers make sense. It is useful for NCEA and university statistics study, workplace quality control, sports and gambling odds, and medical or scientific decision-making, though results are only as reliable as the probabilities and assumptions you enter.
Combinations (order does not matter): How many different groups of 3 can be chosen from 10 people? C(10,3) = 10!/(3! x 7!) = 120 different groups. The NZ Lotto 6/40 has C(40,6) = 3,838,380 possible combinations.
What is probability?
Probability is a number between 0 and 1 (or 0% and 100%) that measures how likely an event is to occur. A probability of 0 means the event is impossible. A probability of 1 means it is certain. A probability of 0.5 means the event occurs half the time. Probabilities are always calculated as: number of ways the event can happen, divided by the total number of equally likely outcomes.
Types of probability
Classical probability applies when all outcomes are equally likely, such as rolling a fair die or drawing a card from a shuffled deck. Empirical probability is based on observed data: if a factory produced 12 defective items out of 1,000, the empirical probability of a defect is 1.2%. Subjective probability is an educated estimate based on judgment, used in forecasting and risk assessment.
The addition rule (OR)
P(A or B) = P(A) + P(B) - P(A and B). The reason we subtract P(A and B) is to avoid counting the overlap twice. If events are mutually exclusive (they cannot both happen at the same time), then P(A and B) = 0 and the formula simplifies to P(A or B) = P(A) + P(B).
The multiplication rule (AND)
For two independent events (where one does not affect the other): P(A and B) = P(A) x P(B). For dependent events: P(A and B) = P(A) x P(B|A), where P(B|A) is the conditional probability of B given that A has occurred. Independence is a crucial assumption: rolling two dice is independent (first roll does not affect second), but drawing cards without replacement is dependent (removing a card changes the remaining pool).
The normal distribution and the 68-95-99.7 rule
The normal distribution (bell curve) is the most important probability distribution in statistics because many real-world measurements naturally follow it: heights, exam scores, measurement errors, financial returns. The 68-95-99.7 rule states: approximately 68% of values fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three. If exam scores have a mean of 65 and standard deviation of 10, then 95% of students score between 45 and 85 (mean plus or minus two standard deviations).
Bayes' theorem and the base rate fallacy
Bayes' theorem is the mathematically correct way to update a probability when new evidence arrives. The base rate fallacy is the common mistake of ignoring how rare an event is when interpreting a positive test result. A medical test that is 99% accurate sounds very reliable, but if the disease it tests for affects only 1% of the population, then a positive result only means about a 17% chance of actually having the disease. The high false positive rate among healthy people (who are 99% of the population) overwhelms the accurate positives. Bayes' theorem is used in medical diagnosis, spam email filtering, financial risk models, and machine learning classification.