The medical test paradox, and redesigning Bayes' rule
by Grant Sanderson · 3Blue1Brown
Follow-up covering why a positive medical test often still means low disease probability, and reframing Bayes' rule in odds form with likelihood ratios (Bayes factors) so an update becomes a single multiplication. It shows the formula most people learn is the wrong parameterisation, and hands over the odds form practitioners actually use.
More resources on Bayes’ Theorem
probabilitycourse.com
ProbabilityCourse.com is an online learning resource that provides a structured probability course with clear explanations, worked examples, and practice problems covering topics from basics to advanced theory.
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Brilliant.org - Bayes' Theorem
A Brilliant wiki page deriving Bayes' theorem from conditional probability, visualising it with Venn diagrams, and working through the two-children paradox, false-positive disease testing, and biased-coin and ball-drawing problems. Readers will be able to set up and solve basic posterior probability questions.
Khan Academy - Bayes' Theorem
A Khan Academy lesson from the conditional probability and independence unit of its free statistics and probability course. It shows how to reverse a conditional probability using Bayes' theorem, so learners can compute the probability of a cause given observed evidence.
Bayes Theorem, the geometry of changing beliefs
Grant Sanderson builds Bayes' theorem from a diagram of overlapping populations rather than the formula, using the medical-test and Steve-the-librarian examples. Viewers finish able to reason about how evidence should shift a prior probability.
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First part of Rasmus Bååth's video introduction to Bayesian data analysis, explaining the approach conceptually before the math. Viewers see how prior information, observed data, and a generative model combine to produce a posterior distribution.