Calculate probabilities for single events, two events, Bayes' theorem, binomial distribution, normal distribution, and combinations
Probability is the mathematical language of uncertainty — a number between 0 and 1 (or 0% and 100%) that quantifies how likely an event is to occur. Whether you are analyzing the odds of winning a game, interpreting a medical test result, estimating quality defects on a production line, or studying statistics for an exam, understanding probability is an essential skill. Our free probability calculator brings six powerful calculation modes into one easy-to-use tool, covering everything from the simplest coin-flip scenario to advanced Bayesian inference and continuous normal distributions. The Basic mode calculates the probability of a single event using the classic formula P(E) = favorable outcomes ÷ total outcomes. If you roll a standard die and want to know the probability of getting a 3, you have 1 favorable outcome out of 6 total, giving P(3) = 1/6 ≈ 16.67%. The result is shown simultaneously as a decimal, percentage, fraction, and odds ratio — letting you communicate probability in whichever format best suits your audience. The Events mode handles two or three simultaneous events A, B, and optionally C. In the independent events sub-mode, the calculator computes the intersection P(A∩B) = P(A) × P(B), union P(A∪B), exclusive-OR P(AΔB), and the probability that neither event occurs. When a third event C is provided, the tool extends all calculations to three events using inclusion-exclusion: P(A∩B∩C) = P(A)×P(B)×P(C), P(A∪B∪C) = P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C). A useful bonus feature calculates the at-least-once probability. The Bayes mode implements Bayes' Theorem in its expanded form. You provide the prior probability P(A), the sensitivity P(B|A), and the false positive rate P(B|¬A). The calculator derives the marginal probability P(B) and computes the posterior probability P(A|B). This is used in medical diagnosis, spam filtering, drug testing, fraud detection, and countless other Bayesian inference problems. The Binomial mode is ideal for repeated-trial experiments where each trial has two outcomes. You enter the number of trials n, success probability p, and target number of successes k, then receive the exact probability P(X=k) plus all cumulative probabilities. An animated PMF bar chart highlights your target bar. The Normal Distribution mode computes P(a ≤ X ≤ b) for any normal distribution with mean μ and standard deviation σ. The bounds support −∞ and +∞ for left-tail and right-tail areas. The CDF is computed using the Abramowitz & Stegun error function approximation, which is accurate to 1.5 × 10⁻⁷. Results include both Z-scores and an interactive SVG bell curve with the probability area shaded. The Combinations & Permutations mode helps you count the number of ways to select or arrange items. Combinations C(n,r) count selections where order does not matter, while permutations P(n,r) count arrangements where order matters.
Understanding Probability
What Is Probability?
Probability is a numerical measure of the likelihood that a specific event will occur, expressed as a value between 0 (impossible) and 1 (certain). A probability of 0.5 means the event is equally likely to occur or not. In classical probability theory, for equally likely outcomes, P(E) = n(E) / n(S), where n(E) is the number of favorable outcomes and n(S) is the total number of outcomes in the sample space S. Probability can also be expressed as a percentage (0–100%), a fraction (1/6), or odds (5 to 1 against). Theoretical probability assumes ideal conditions, while empirical probability is based on observed frequencies from experiments.
How Is Probability Calculated?
The calculation method depends on the type of problem. For single events: P(E) = favorable / total. For two independent events, P(A∩B) = P(A) × P(B); P(A∪B) = P(A) + P(B) − P(A∩B). For three independent events, P(A∩B∩C) = P(A)×P(B)×P(C); P(A∪B∪C) via inclusion-exclusion. For Bayes' theorem: P(A|B) = P(B|A) × P(A) / [P(B|A) × P(A) + P(B|¬A) × P(¬A)]. For normal distribution: P(a ≤ X ≤ b) = Φ((b−μ)/σ) − Φ((a−μ)/σ) where Φ is the standard normal CDF. For binomial: P(X=k) = C(n,k) × pᵏ × (1−p)^(n−k).
Why Does Probability Matter?
Probability is fundamental to virtually every field of human knowledge and decision-making. In medicine, Bayesian probability helps doctors interpret test results — a positive test does not necessarily mean high probability of disease if the prior prevalence is low (the base rate fallacy). In finance, probability theory underlies risk management, option pricing (Black-Scholes model), and actuarial calculations. In engineering, binomial distribution models defect rates and quality control. In machine learning, probability is used in Naïve Bayes classifiers, logistic regression, and neural network outputs. The normal distribution describes IQ scores, measurement errors, and countless natural phenomena.
限制和假设
Each probability model rests on assumptions that may not hold in practice. Classical probability assumes equally likely outcomes — not valid for a loaded die. Independence assumptions fail when events are correlated. The binomial distribution requires fixed n, constant p, and independent trials. The normal distribution is a continuous approximation — real data may be skewed or heavy-tailed. The three-event extension in Events mode assumes independence for all pairs involving C; if C is dependent on A or B, provide the known joint probability manually. Factorial calculations exceed 64-bit precision for n > 170. Always verify that your inputs sum to valid probability constraints (0 ≤ P ≤ 1).
How to Use the Probability Calculator
选择计算模式
Select one of the six tabs at the top: Basic (single event), Events (two or three events A, B, C), Bayes (conditional/Bayesian), Binomial (repeated trials), Normal (continuous distribution area), or Combinations (counting). Each mode has its own inputs tailored to that type of probability problem.
输入您的数值
Type your probability values in the input fields. You can enter probabilities as decimals (0.25), percentages (25%), or fractions (1/4) — the calculator auto-detects the format. For Normal mode, enter the mean μ, standard deviation σ, and bounds (use -inf or inf for infinite tails). Use the Quick Presets buttons for instant real-world examples.
阅读您的结果
Results appear instantly as you type. The main probability is shown prominently with a visual chart. In Normal mode, the shaded bell curve shows the exact probability area. In Events mode, the Venn diagram updates live as you change P(A), P(B), and P(C). For Bayes mode, the prior-vs-posterior bar shows how much the evidence shifted the probability.
Review Step-by-Step Solutions
Scroll down in the results card to find the step-by-step formula derivation. This shows exactly how the calculator arrived at the answer using your specific numbers — including Z-score conversion for normal distribution problems. Use Export CSV to download all calculated values for reports or further analysis.
常见问题
What is the difference between independent and dependent events?
Two events are independent if the occurrence of one does not affect the probability of the other. For example, flipping a coin twice — the first flip does not change the probability of the second flip. For independent events, P(A∩B) = P(A) × P(B). Dependent events are linked — the outcome of one affects the other. For example, drawing two cards from a deck without replacement: the probability of the second draw changes depending on the first. For dependent events, P(A∩B) = P(A) × P(B|A), where P(B|A) is the conditional probability of B given A. The Events tab lets you choose between these two modes and also enter a known joint probability.
How does the Normal Distribution tab work?
The Normal tab computes the area under a normal curve between two bounds. Enter the mean μ (center), standard deviation σ (spread), and the lower and upper bounds of the region. For a left-tail area (P(X < b)), enter -inf as the lower bound. For a right-tail area (P(X > a)), enter inf as the upper bound. The calculator converts each bound to a Z-score: Z = (X − μ) / σ, then uses the standard normal CDF Φ(Z) to find the area. The probability P = Φ(Z₂) − Φ(Z₁). An SVG bell curve is drawn with the probability region shaded in real time. The CDF uses the Abramowitz and Stegun error function approximation, accurate to 1.5 × 10⁻⁷.
How do I calculate probabilities for three events A, B, and C?
In the Events tab, enter probabilities for P(A) and P(B), then optionally fill in P(C). When P(C) is provided, the calculator computes P(A∩B∩C) = P(A)×P(B)×P(C) and P(A∪B∪C) using the full inclusion-exclusion formula: P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C). It also shows the probability that exactly one event occurs and that none occur. Note that this three-event extension assumes all three events are mutually independent. If you have known joint probabilities for pairs, you would need to enter those manually.
What is Bayes' theorem and why is it surprising?
Bayes' theorem updates the probability of a hypothesis A after observing evidence B. The formula is P(A|B) = P(B|A) × P(A) / P(B). What surprises most people is how dramatically the base rate (prior) affects the posterior. Consider a disease affecting 1% of the population, with a test that is 99% sensitive and 5% specific. A positive test result still means only about 17% chance of having the disease — because false positives among the large healthy population overwhelm the true positives from the rare sick population. This is the base-rate fallacy, and it explains why medical screening of low-prevalence conditions is tricky to interpret without Bayesian reasoning.
When should I use the binomial distribution versus the normal distribution?
Use the binomial distribution when you have discrete, countable outcomes: a fixed number of trials n, each with exactly two possible outcomes, and constant success probability p. Examples: coin flips, quality defects in a batch, clinical trial successes. Use the normal distribution when your variable is continuous — heights, weights, test scores, measurement errors — or when your discrete data is approximately normally distributed. When n is large and p is moderate (np ≥ 5 and n(1−p) ≥ 5), the binomial distribution can be approximated by a normal distribution with μ = np and σ = √(np(1−p)). The binomial tab on this calculator tells you whether this approximation is valid for your specific parameters.
Can I enter probabilities as percentages or fractions?
Yes. All probability input fields on this calculator accept three formats: decimal (e.g. 0.1667), percentage (e.g. 16.67%), or fraction (e.g. 1/6). The calculator auto-detects the format — if you enter a value greater than 1 but ≤ 100, it interprets it as a percentage. If you use a slash, it treats the input as a fraction. This makes it easy to enter probabilities like 1/52 for drawing a specific card from a deck, 3/4 for a 75% probability, or 0.01 for a 1% prevalence rate. Normal distribution bound inputs accept plain real numbers and also -inf / inf for infinite tails.