Probability Calculator

How to use the probability calculator

1. Pick a type of problem in Calculation: Single event (favourable ÷ possible), Complement (not A), Two events (A and B, A or B), Conditional probability (Bayes’ theorem), At least one success in n trials, Binomial distribution, Odds to probability, or Probability to odds.
2. Fill in the fields. Probabilities can be typed as a fraction (1/6), a decimal (0.25) or a percentage (25%). Counts such as n – number of trials must be whole numbers.
3. For two events, choose the Relationship between A and B: Independent, Mutually exclusive, or I know P(A and B). For odds, say whether they are Odds in favour (for) or Odds against.
4. Results update as you type (the Calculate button does the same). Each result is shown as an exact Fraction, a Decimal and a Percent, and Step by step shows the formula with your numbers. Copy copies the result table as text.

The tool works with exact fractions: 1/6 stays 1/6 instead of becoming 0.1666…, so the answer to “at least one six in four rolls” is exactly 671/1296. When a fraction gets too long to read, only the decimal is shown.

The formulas used

• Single event: P(A) = favourable outcomes ÷ possible outcomes, when all outcomes are equally likely.
• Complement: P(not A) = 1 − P(A).
• Independent events: P(A and B) = P(A) × P(B).
• Mutually exclusive events: P(A and B) = 0, so P(A or B) = P(A) + P(B).
• Addition rule (any events): P(A or B) = P(A) + P(B) − P(A and B).
• Conditional probability: P(A | B) = P(A and B) ÷ P(B).
• Bayes’ theorem: P(A | B) = P(B | A) × P(A) ÷ P(B), with P(B) = P(B | A) × P(A) + P(B | not A) × P(not A).
• At least one success: 1 − (1 − p)ⁿ.
• Binomial distribution: P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, with mean np and variance np(1 − p).
• Odds: odds for a:b give P = a ÷ (a + b); odds against a:b give P = b ÷ (a + b).

Bayes’ theorem and the 2 × 2 table

The Bayes mode is built for questions like “a test is 95% sensitive with a 5% false positive rate, and 1% of people have the condition: what is the chance that a positive result is a true positive?” Enter P(A) – prior (base rate) = 1%, P(B | A) – true positive rate = 95% and P(B | not A) – false positive rate = 5%. The answer, P(A | B) = 19/118 ≈ 16.1%, surprises most people: because the condition is rare, false positives outnumber true ones. The 2 × 2 table of joint probabilities shows all four combinations (A and B, A and not B, not A and B, not A and not B) with their totals, and the steps restate it as natural frequencies out of 10,000 people.

Binomial results and large numbers

The binomial mode gives P(X = k), P(X < k), P(X ≤ k), P(X > k) and P(X ≥ k) together, plus the mean, variance and standard deviation. Up to 1,000 trials every value is computed exactly with big integers. From 1,001 to 100,000 trials the tool switches to floating-point arithmetic in log space and says so in the steps; the results are then accurate approximations. The “at least one success” mode accepts up to 1,000,000 trials and also tells you how many trials give a 50% and a 95% chance of at least one success.