Probability Calculator

Independent events do not affect each other's probability — flipping a coin and rolling a die are independent because the coin result does not change the die odds. Mutually exclusive events cannot happen at the same time — drawing one card cannot be both a heart and a spade. The key difference: independent events have P(A∩B) = P(A) × P(B), while mutually exclusive events have P(A∩B) = 0. Independent events can definitely co-occur; mutually exclusive events cannot co-occur at all.

Conditional probability P(A|B) answers: "given that B happened, how likely is A now?" For example, the probability of having a disease (A) given a positive test result (B) is a conditional probability. This is fundamentally different from the unconditional probability P(A) of having the disease in the general population. Bayes' theorem mathematically relates P(A|B) to P(B|A) and is how email spam filters learn to classify messages, how medical tests are interpreted, and how recommendation systems predict what you will like.

Probability is used everywhere: insurance companies set premiums based on claim probabilities; weather forecasts report "70% chance of rain" using probabilistic models; financial markets use probability to price options and manage risk; quality control uses sampling probabilities to accept or reject production batches; A/B testing uses p-values to decide if a website change is effective; machine learning classifiers output probability scores for predictions. Even your phone's keyboard uses probability to predict the next word you will type.

P(A∩B) — the joint probability of A and B — is the overlapping region where two circles intersect in a Venn diagram. The left circle represents all outcomes where A occurs, the right circle represents all where B occurs, and the overlap is where both A and B occur simultaneously. The size of this overlap relative to the total diagram area represents the probability. If the circles do not touch at all (mutually exclusive), P(A∩B) = 0. If one circle is entirely inside the other, P(A∩B) = P(the smaller event).

The law of large numbers states that as you repeat a random experiment more times, the average result gets closer to the expected value. A coin flipped 10 times might show 70% heads, but after 10,000 flips it will be very close to 50%. This is why casinos always win in the long run despite individual gamblers hitting jackpots, why insurance companies can profit despite unpredictable individual claims, and why large sample sizes give more reliable statistical results. Small samples have high variance; large samples converge to the true probability.