Bayes Theorem Calculator Formula

Understand the math behind the bayes theorem calculator. Each variable explained with a worked example.

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Formulas Used

P(A|B) - Posterior

posterior = (p_b_given_a * p_a) / p_b

Posterior (%)

posterior_pct = ((p_b_given_a * p_a) / p_b) * 100

P(B) - Total Evidence

total_evidence = p_b

Variables

VariableDescriptionDefault
p_aP(A) - Prior0.01
p_b_given_aP(B|A) - Likelihood0.9
p_b_given_not_aP(B|not A) - False Positive Rate0.05
p_bDerived value= p_b_given_a * p_a + p_b_given_not_a * (1 - p_a)calculated

How It Works

How to Apply Bayes' Theorem

Formula

P(A|B) = P(B|A) * P(A) / P(B)

where P(B) = P(B|A)*P(A) + P(B|not A)*P(not A)

Bayes' theorem updates a prior belief P(A) after observing evidence B. The likelihood P(B|A) measures how probable the evidence is if A is true. The denominator P(B) normalizes the result.

Worked Example

A disease affects 1% of the population. A test is 90% sensitive and has a 5% false positive rate. If someone tests positive, what is the probability they have the disease?

p_a = 0.01p_b_given_a = 0.9p_b_given_not_a = 0.05
  1. 01P(A) = 0.01 (prior: disease prevalence)
  2. 02P(B|A) = 0.9 (sensitivity)
  3. 03P(B|not A) = 0.05 (false positive rate)
  4. 04P(B) = 0.9 * 0.01 + 0.05 * 0.99 = 0.009 + 0.0495 = 0.0585
  5. 05P(A|B) = (0.9 * 0.01) / 0.0585 = 0.009 / 0.0585 ≈ 0.1538
  6. 06Despite a positive test, there is only about a 15.4% chance of having the disease.

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