What is Bayes' theorem and how is it used here?

Bayes' theorem provides a formula for computing conditional probability from three known values: the prior probability P(A), the true positive rate P(B|A), and the false positive rate P(B|not A). The formula is: P(A|B) = P(B|A) x P(A) / [P(B|A) x P(A) + P(B|not A) x P(not A)]. The denominator is the total probability of B, often written P(B). Bayes' theorem is widely used in medicine, spam filtering, machine learning, and forensics.

Why does a positive test result not always mean you have the condition?

Even a highly accurate test can produce a misleading result when the underlying condition is rare. If only 1 in 1,000 people have a disease, a test with a 99% true positive rate and a 5% false positive rate will still generate about 50 false positives for every true positive in that population. This is why doctors consider the prevalence of a condition alongside test accuracy when interpreting results. The conditional probability formula captures this relationship precisely.

Conditional Probability Calculator

This calculator works out conditional probability - the chance that event A is true given that you have observed evidence B - using Bayes' theorem. It suits situations like interpreting a medical test result, a spam filter flag, or a quality control failure, where one piece of evidence needs weighing against how common the underlying event actually is. You enter three percentages: the prior probability P(A), how likely A is before you see any evidence, such as how common a disease is in the population tested; the true positive rate P(B|A), how often the evidence appears when A is true; and the false positive rate P(B|not A), how often the evidence appears even when A is false. From these three inputs the calculator instantly returns P(A|B), the updated probability of A now that B has occurred, alongside P(A|not B), the total probability of the evidence P(B), and P(not A|B). A calculation breakdown shows every intermediate step, including the joint probabilities P(A and B) and P(not A and B), plus a worked example using the default figures so you can check the maths. A plain-English verdict below the results explains whether the evidence has raised or lowered the probability of A, and by how much. Results are mathematically exact for the probabilities you supply, so make sure your estimates are realistic before relying on the output.

Calculate.co.nz is proud to be partnered with Premium Homes, a recognised leader in eco-friendly, sustainable, and energy-efficient homebuilding. With a dedicated team and award-winning experience, they create homes that prioritise health, comfort, and long-term performance. Their founders, Andrew and Kelly, set out to raise the standard of residential construction in New Zealand by combining practical building expertise with a clear commitment to doing things better for homeowners.

Standard formula Bayes' theorem: P(A|B) = P(B|A) x P(A) / P(B). Results are exact given the inputs you provide.

1. Prior and Evidence Rates

Prior probability of A, P(A) e.g. disease prevalence

Enter a value between 0 and 100.

True positive rate P(B|A) sensitivity / hit rate

Enter a value between 0 and 100.

False positive rate P(B|not A) 1 minus specificity

Enter a value between 0 and 100.

2. What the Inputs Mean

Use a medical test as an example:

P(A): how common is the condition in the population being tested?
P(B|A): if a person has the condition, how often does the test correctly come back positive?
P(B|not A): if a person does not have the condition, how often does the test incorrectly come back positive?

The calculator works for any A and B, not just medical tests.

Conditional Probability Results

P(A|B)
-
Prob. of A given B occurred

P(A|not B)

Prob. of A given B did not occur

P(B)

Total probability of B

P(not A|B)

Prob. A did not occur given B

Calculation Breakdown

Prior P(A)-

Prior P(not A)-

True positive rate P(B|A)-

False positive rate P(B|not A)-

P(A and B) = P(B|A) x P(A)-

P(not A and B) = P(B|not A) x P(not A)-

P(B) = P(A and B) + P(not A and B)-

P(A|B) = P(A and B) / P(B)-

Worked Example Check

With the default inputs (P(A)=30%, P(B|A)=70%, P(B|not A)=20%):

P(A and B)0.70 x 0.30 = 0.2100

P(not A and B)0.20 x 0.70 = 0.1400

P(B)0.2100 + 0.1400 = 0.3500

P(A|B)0.2100 / 0.3500 = 60.00%

Even though A has only a 30% prior probability, the positive evidence of B raises it to 60%. This is the core insight of Bayesian reasoning.

Result: Enter your values above to see the conditional probability.

What Is Conditional Probability?

Conditional probability is the probability that event A will occur, given that event B has already occurred. It is written P(A|B), which is read as "the probability of A given B". The vertical bar means "given that".

The basic formula is:

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

If you know P(A), P(B|A) and P(B|not A), you can use Bayes' theorem to compute P(A|B) without knowing P(A and B) directly.

Bayes' Theorem

Bayes' theorem expands the conditional probability formula so you can compute P(A|B) from three quantities you typically do know:

P(A|B) = P(B|A) x P(A) / [ P(B|A) x P(A) + P(B|not A) x P(not A) ]

The numerator is P(A and B). The denominator is the total probability of B, using the law of total probability: P(B) = P(B|A) x P(A) + P(B|not A) x P(not A).

Step-by-Step Method

Identify the prior probability P(A): how likely is A before observing B?
Identify the true positive rate P(B|A): if A is true, how often is B observed?
Identify the false positive rate P(B|not A): if A is false, how often is B still observed?
Calculate P(A and B) = P(B|A) x P(A).
Calculate P(not A and B) = P(B|not A) x P(not A), where P(not A) = 1 - P(A).
Calculate total P(B) = P(A and B) + P(not A and B).
Divide: P(A|B) = P(A and B) / P(B).

Worked Example (Default Values)

Step	Calculation	Value
Prior P(A)	Given	0.30
Prior P(not A)	1 - 0.30	0.70
True positive P(B\|A)	Given	0.70
False positive P(B\|not A)	Given	0.20
P(A and B)	0.70 x 0.30	0.2100
P(not A and B)	0.20 x 0.70	0.1400
P(B)	0.2100 + 0.1400	0.3500
P(A\|B)	0.2100 / 0.3500	0.6000 (60.00%)

Why This Matters: The Base Rate Effect

Conditional probability is counterintuitive when the base rate (prior probability) is very different from 50%. Consider a rare disease with a prevalence of 1% in the tested population. A test with a 99% true positive rate and a 5% false positive rate gives:

P(A and B) = 0.99 x 0.01 = 0.0099
P(not A and B) = 0.05 x 0.99 = 0.0495
P(B) = 0.0099 + 0.0495 = 0.0594
P(disease | positive test) = 0.0099 / 0.0594 = 16.7%

Even with a highly accurate test, a positive result in a low-prevalence population still has an 83% chance of being a false alarm. This is why screening programmes are targeted at higher-risk groups, where the prior probability is higher.

Common Applications

Medical testing: interpreting a positive or negative diagnostic result given test accuracy and disease prevalence.
Spam filters: estimating the probability that an email is spam given that it contains certain words.
Quality control: determining the probability that a defect is from a specific production line given an observed failure.
Risk assessment: updating the probability of a financial, engineering, or safety event as new data arrives.
Machine learning: naive Bayes classifiers use this formula to categorise data.

P(A|not B): Probability Given No Evidence

This calculator also shows P(A|not B), the probability of A given that B did not occur. This uses the same Bayes structure with P(not B|A) = 1 - P(B|A) and P(not B|not A) = 1 - P(B|not A). A negative result updates the prior downward by a factor that depends on how reliable the test is at ruling out A.

Related Calculators

Maths and Stats Calculators: full hub for probability, statistics and maths tools.
Bayes Theorem Calculator: direct P(A|B) entry using raw probabilities.
Probability Calculator: basic probability, odds and event combinations.
Coin Flip Probability Calculator: binomial probability for repeated trials.
Combination (nCr) Calculator: count combinations for probability problems.

Sources and method: Bayes' theorem from T. Bayes (1763), published posthumously in Philosophical Transactions of the Royal Society. Law of total probability: standard probability theory. Formula reference: DeGroot and Schervish, Probability and Statistics, 4th edition.

Results are mathematically exact given the input probabilities you supply. The calculator assumes the inputs are independent and that A and B are well-defined mutually understood events. It does not perform statistical inference from raw data.