Unit 17
Conditional probability.
How new information changes a chance: "given that", independent events, probability trees, and Bayes' theorem.
Before this unit
The unit, lesson by lesson
Lessons
4 lessons · 20 practice questions · about 2 hours
Given that
What information does to a chance: count again inside the event you know happened, $P(A \mid B)$ against $P(B \mid A)$, independent events, and why mutually exclusive events are not independent.
Trees with conditions
A tree with a conditional chance on every branch: split a hall of $1000$ people into ill and healthy, then by test result, multiply along a path, add paths, and draw marbles without putting back.
Base rates
Why a test that is $90\%$ accurate is wrong about most of its positives when the disease is rare: count the false positives, find when a positive is half and half, and see what a better test does and does not do.
Bayes' theorem
From counting in the hall to a formula: Bayes' theorem turns the condition round, a retest beats the base rate, what comes out after one piece of evidence is the starting point for the next, and the prosecutor's fallacy.