Probability Solver

The Probability Solver handles the questions where the hard part is setting up the count. Permutations and combinations with and without repetition. Probability of unions, intersections, and complements. Conditional probability and independence. Bayes' theorem with the tree or table written out. Expected value and variance of a discrete random variable. Binomial, geometric, hypergeometric, Poisson, and normal distribution probabilities, including at-least and at-most questions. It states the sample space or the counting model before computing anything. Problems involving two dice, cards, arrangements with restrictions, and the birthday-style complement argument are set up with the outcome count written before the probability.

Type the question in full or photograph it: "probability of at least two heads in five tosses", "a bag has 4 red and 6 blue, draw 3 without replacement, P(exactly 2 red)", "P(A|B) given P(A)=0.3, P(B)=0.5, P(A and B)=0.1". Say whether draws are with or without replacement and whether order matters if the problem does not make it clear. Set Answer form to exact for fractions, decimal for a rounded probability, or both. For a distribution question, give the parameters, such as n and p for binomial or the mean for Poisson, and the value or range being asked about. For a two-way table or a Venn diagram problem, paste the table or list the region counts and name the events.

Output: the probability first, as a fraction and a decimal, then the model: the sample space size or the distribution named, the events defined, the formula, the substitution, and the arithmetic. Bayes problems show the tree with each branch probability. The working is shown so you can check it. For data sets, hypothesis tests, and confidence intervals, use the Statistics Solver. For pure counting without probability, the Number Theory Solver also handles combinatorics. A follow-up can rerun the problem with replacement instead of without, or ask for the full probability distribution table rather than one value. Independence checks and the multiplication rule are written explicitly at each branch.