Probability Calculator

Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). The classical definition applies when all outcomes are equally likely: P(A) = (number of favorable outcomes) ÷ (total number of outcomes). A fair coin has two equally likely faces, so P(heads) = 1/2 = 0.5 = 50%. A standard six-sided die has six equally likely faces, so P(rolling a 4) = 1/6 ≈ 16.67%. This calculator accepts favorable and total counts and returns the probability as a simplified fraction, decimal, and percentage simultaneously.

The complement rule states that an event either happens or it does not: P(not A) = 1 − P(A). If the chance of rain is 30%, the chance of no rain is 70%. Complements are essential when direct counting is awkward — it is often easier to count failures than successes. Enter any valid probability between 0 and 1 (or as a percent) and the tool computes its complement instantly.

For two independent events A and B, the joint probability P(A and B) equals P(A) × P(B). Independence means one outcome does not influence the other — two coin flips, two dice rolls, or drawing with replacement. The union rule for any two events is P(A or B) = P(A) + P(B) − P(A and B), which avoids double-counting outcomes that satisfy both events. Conditional probability P(A|B) = P(A and B) / P(B) updates your belief about A after learning B occurred.

Classic dice example: when rolling two fair dice, there are 36 equally likely ordered pairs. Exactly six pairs sum to 7 — (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — so P(sum = 7) = 6/36 = 1/6 ≈ 16.67%. Card problems, lottery odds, and genetics Punnett squares all use the same counting logic. Always verify that outcomes are equally likely before applying the favorable-over-total formula; weighted spinners and biased coins require different models.

Results appear as simplified fractions (using the fraction calculator logic), exact decimals, and rounded percentages. Use single-event mode for straightforward counting problems, complement mode for "at least one" scenarios via 1 − P(none), and two-event mode when combining independent probabilities. For random simulations and repeated trials, pair this tool with the random number generator to model experiments numerically.

Probability underpins statistics, risk assessment, game design, and machine learning. Whether you are checking homework on coin flips, estimating inspection defect rates, or explaining why a 90% success rate repeated twice is not 180%, this calculator keeps the arithmetic transparent with step-by-step working shown below each result.