Free Probability Calculator
Calculate the probability of single and combined events. Computes P(A), P(not A), P(A and B), and P(A or B) using classical probability rules. Covers complement, multiplication, and addition rules. Free, private — all calculations run in your browser.
Probability Calculator — Events, Binomial & Combinations
Calculate event probabilities, binomial distribution outcomes, and permutation/combination counts. Supports independent events (P(A∪B), P(A∩B)), exact binomial probability P(X=k), cumulative P(X≤k), and combinatorial counting. Used in statistics, data science, gaming and finance.
How to Use This Calculator
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Choose a mode: Basic Probability, Binomial Distribution, or Permutations & Combinations.
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For Basic mode: enter two probabilities between 0 and 1.
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For Binomial: enter n (trials), k (successes wanted), and p (probability of success per trial).
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For Perm/Comb: enter n (total items) and r (items chosen).
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Results appear instantly. Use "Export PDF" to save.
📐 How This Is Calculated
P(A∪B) = P(A)+P(B)−P(A)P(B) | Binomial: P(X=k) = C(n,k)×pᵏ×(1−p)ⁿ⁻ᵏ
P(A)—Probability of event A occurring (0 to 1)n—Number of independent trials in binomial distributionk—Number of successes we are calculating the probability forp—Probability of success on a single trialC(n,k)—Combination — number of ways to choose k items from nReference: Probability theory — Kolmogorov axioms; Blaise Pascal & Pierre de Fermat (1654)
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Results are provided for informational and educational purposes only. They should not be used as a substitute for professional financial, engineering, medical, or legal advice. Always verify outputs with a qualified professional before making important decisions. Roughtools makes no warranties regarding accuracy or completeness for your specific situation.
About This Probability Calculator
The Probability Calculator computes the likelihood of events using the rules of classical probability. Probability is the mathematical language of uncertainty — it quantifies how likely an event is to occur on a scale from 0 (impossible) to 1 (certain). This calculator covers the foundational rules that apply whenever outcomes are equally likely and events are independent. It is ideal for statistics coursework, everyday decision-making under uncertainty, and understanding the odds in games and risk scenarios.
The Core Probability Formulas
P(not A) = 1 − P(A) [complement rule]
P(A and B) = P(A) × P(B) [independent events]
P(A or B) = P(A) + P(B) − P(A and B) [addition rule]
Types of Probability
Classical probability applies when all outcomes are equally likely — rolling a fair die, drawing a card from a shuffled deck. P(A) is the count of favorable outcomes divided by the total number of possible outcomes. Empirical probability is based on observed data from experiments: if a machine produces 7 defective parts out of 200, the empirical probability of a defect is 7/200 = 3.5%. Subjective probability is an expert estimate when no frequency data exists — a doctor's assessment that a treatment has a 70% chance of success based on clinical experience.
Independent vs. Dependent Events
Two events are independent if knowing the outcome of one gives no information about the other. Coin flips are independent — the coin has no memory. In this case, P(A and B) = P(A) × P(B). Events are dependent if the outcome of one changes the probability of the other. Drawing cards without replacement is dependent — the remaining deck composition changes after each draw. For dependent events, use conditional probability: P(A and B) = P(A) × P(B|A), where P(B|A) is the probability of B given A.
Complementary Events
The complement of event A (written "not A" or A') includes every outcome where A does NOT occur. Since every outcome is either A or not A: P(A) + P(not A) = 1, therefore P(not A) = 1 − P(A). The complement rule is especially useful for "at least one" problems. For example: "What is the probability of rolling at least one six in four rolls of a die?" Rather than summing many terms, compute P(no sixes) = (5/6)⁴ ≈ 0.482, so P(at least one six) = 1 − 0.482 ≈ 0.518.
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When to Use This Calculator
Calculate the probability of drawing specific cards, rolling certain dice combinations, or winning a hand. Make decisions based on actual odds rather than gut feeling.
Understand the probability of risk events — flood, accident, illness — over a given time period. Compare risk probabilities to make informed coverage decisions.
Verify probability calculations for class assignments. Check complement rule, addition rule, and multiplication rule answers step by step.
Model the probability that multiple independent risk factors occur simultaneously. Estimate the chance a project encounters at least one major obstacle during development.
Calculate the probability of offspring inheriting specific genetic traits using Mendelian genetics. Model dominant and recessive allele combinations with the multiplication rule.
💡 Pro Tips
Use the complement rule when "at least one" problems seem hard. Instead of calculating P(at least one success in n trials) directly — which requires summing many terms — calculate P(zero successes) and subtract from 1: P(at least one) = 1 − P(none). For example, probability of rolling at least one 6 in four dice rolls = 1 − (5/6)⁴ ≈ 0.518.
Apply the multiplication rule only for truly independent events. P(A and B) = P(A) × P(B) assumes knowing A occurred tells you nothing about whether B occurs. In practice, events are often dependent — drawing cards without replacement, conditional medical test results, correlated market returns. Always verify independence before multiplying.
Sample size dramatically affects empirical probability accuracy. If you observe 6 heads in 10 flips, the empirical probability is 0.6 — but the true probability may still be 0.5. With 1,000 flips, empirical probability converges much closer to the true value. Empirical probability estimates improve as sample size grows, following the Law of Large Numbers.
Probability quantifies uncertainty — it does not guarantee outcomes. A 90% probability event fails 1 in 10 times. A 1% probability event still occurs regularly over many trials. Misunderstanding probability as certainty is called the "gamblers' fallacy." Past outcomes of independent random events do not affect future probabilities.
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