Bayes Theorem Calculator

Understanding Bayes' Theorem and Posterior Probability

Bayes' Theorem is a fundamental principle of probability theory and statistics named after Thomas Bayes. It describes how to update the probability of a hypothesis ($A$) when new evidence ($B$) becomes available.

Bayes' Theorem Formula

The mathematical formula for Bayes' Theorem is expressed as:

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

Expanding the total probability $P(B)$ via the law of total probability:

$$P(B) = P(B|A) \cdot P(A) + P(B|\neg A) \cdot P(\neg A)$$

Key terms in the equation:

• $P(A|B)$ (Posterior Probability): The probability of hypothesis $A$ given evidence $B$.
• $P(A)$ (Prior Probability): The initial probability of $A$ before seeing evidence $B$.
• $P(B|A)$ (Likelihood): The probability of seeing evidence $B$ given hypothesis $A$ is true.
• $P(B|\neg A)$ (False Positive Rate): The probability of seeing evidence $B$ when hypothesis $A$ is false.

Practical Example: Medical Test Interpretation

Suppose a medical condition affects 1% of a population ($P(A) = 0.01$). A screening test has 95% sensitivity ($P(B|A) = 0.95$) and a 5% false positive rate ($P(B|\neg A) = 0.05$). If a patient tests positive ($B$), the actual posterior probability that they have the condition is:

$$P(A|B) = \frac{0.95 \times 0.01}{(0.95 \times 0.01) + (0.05 \times 0.99)} = \frac{0.0095}{0.0095 + 0.0495} \approx 16.10\%$$

For other statistical probability calculations, check our Probability Calculator and Odds Probability Calculator.