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Bayes' Theorem, named after the English statistician Thomas Bayes, is a fundamental concept in probability theory and statistics. It provides a way to update the probability of a hypothesis based on new evidence. The theorem is expressed mathematically as:
[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} ]
Where:
( P(A|B) ) is the posterior probability: the probability of event A occurring given that B is true.
( P(B|A) ) is the likelihood: the probability of event B occurring given that A is true.
( P(A) ) is the prior probability: the initial probability of event A.
( P(B) ) is the marginal probability: the total probability of event B occurring.
Example: Medical Diagnosis
Consider a scenario where a medical test is used to diagnose a disease. The test has a sensitivity (true positive rate) of 90% and a specificity (true negative rate) of 80%. The prevalence of the disease in the population is 5%. We want to find the probability that a person who tests positive actually has the disease.
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