Statistics

Bayes Theorem Calculator

The Bayes Theorem Calculator combines a prior probability P(A), a sensitivity P(B|A), and a false positive rate P(B|not A) to compute the posterior probability P(A|B) as a percentage.

What this calculator does

The Bayes Theorem Calculator estimates your result instantly from the values you enter, using standard statistics formulas. The Bayes Theorem Calculator combines a prior probability P(A), a sensitivity P(B|A), and a false positive rate P(B|not A) to compute the posterior probability P(A|B) as a percentage. Calculations run in your browser, so nothing you type is sent to a server, and the output is an estimate rather than a certified or official figure.

Bayes Theorem Calculator

Result

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P(A|B) — posterior

About Bayes Theorem Calculator

This calculator applies Bayes' theorem to update a prior probability after observing evidence. Enter how likely A is beforehand, how often the evidence appears when A is true, and how often it appears when A is false, and it returns the revised probability of A.

All three inputs are probabilities between 0 and 1, and the result is expressed as a percentage.

Formula & Mathematical Calculation

P(A|B) — posterior = f(P(A) — prior, P(B|A) — sensitivity, P(B|not A) — false positive rate)

Calculated according to standard stats formulas implemented in the calculator core.

Calculation Example

With a prior P(A) = 0.01, sensitivity P(B|A) = 0.9, and false positive rate P(B|not A) = 0.05, the total evidence probability is P(B) = 0.9×0.01 + 0.05×0.99 = 0.009 + 0.0495 = 0.0585.

The posterior is P(A|B) = 0.009 ÷ 0.0585 = 0.1538462, shown as 15.385%. Even with strong evidence, the low prior of 1% keeps the updated probability modest.

Key Benefits

  • Applies Bayes' theorem to update a prior with new evidence
  • Uses the full formula with both true positive and false positive rates
  • Reports the posterior probability directly as a percentage
  • Shows how a low prior probability tempers even strong evidence

How to Use this Calculator

  1. Enter the prior P(A)
    Type the prior probability of A, between 0 and 1, into the P(A) field.
  2. Enter the sensitivity P(B|A)
    Type the probability of the evidence given A into the P(B|A) field.
  3. Enter the false positive rate P(B|not A)
    Type the probability of the evidence when A is false into the P(B|not A) field.
  4. Calculate
    Press Calculate and read the posterior P(A|B) result box.

Assumptions & Limitations

What the calculation assumes

  • Every value you enter is treated as an exact, known amount.
  • Any rate or interval you enter is assumed to hold for the whole period unless the form asks for it separately.
  • Results use standard published formulas and account only for the values entered above.
  • Rounding is applied for display; the underlying calculation uses your full precision.
  • Results are returned in the same units you enter, unless the tool states the output unit.

What it does not cover

  • Bayes Theorem Calculator produces a calculated result for reference and learning, not a certified or official measurement.
  • Real-world outcomes differ when tolerances, rounding rules or local standards apply.
  • Definitions and standards vary by region and change over time; verify the convention that applies to your case.
  • Figures are only as accurate as the inputs you supply — check them before acting on the result.
  • For work where the result carries real consequences, confirm it against an authoritative source.

Frequently Asked Questions

P(A|B) is the updated probability that A is true after taking the observed evidence B into account, combining your prior with how the evidence behaves.

The denominator is the total probability of the evidence, computed as sensitivity times the prior plus the false positive rate times one minus the prior.

When the prior probability of A is very small, false positives from the large complementary group can dominate, keeping the posterior lower than the sensitivity might suggest.

All three probabilities must be between 0 and 1, and the resulting evidence probability must be greater than zero, or the tool shows an error.

Free to use, no account needed. Calculations run in your browser, so nothing you type is sent to a server. Results are estimates — how we build and check our calculators.