When you know Event B happened, how does that change the probability of Event A? Build trees, see how conditioning shrinks the sample space, and test if events are truly independent.
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Scenario
Disease prevalence is 1%. A test has 95% sensitivity and 10% false-positive rate. Despite sounding accurate, a positive result only means ~8.8% chance of actually having the disease.
Probability Tree
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Conditional Probability
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Explanation
When a disease is rare (1% prevalence), most people who test positive are actually healthy, even when the test catches 95% of true cases (its sensitivity; overall accuracy here is about 90%). This is the "base rate fallacy": ignoring how common the condition is.
Step-by-Step
With 1000 people: ~10 have the disease. The test correctly identifies ~9.5 of them (Test+). But of the 990 healthy people, 10% also test positive: that is ~99 false positives. So of ~108 total positives, only ~9 are true positives.
P(Disease | Test+) ≈ 8.8%, not 95%. Rare events require tests with very low false-positive rates to avoid being swamped by false positives.
Tip: Click any tree node above to see how the sample space shrinks.
Sample Space
Independence Checker
Core Concepts
Conditional Probability
P(A|B)
The probability of A given that B has already occurred. Conditioning on B restricts the sample space to only events where B is true.
Multiplication Rule
P(A ∩ B) = P(B) × P(A|B)
Joint probability equals the probability of B times the conditional probability of A given B. This is what each tree path computes.
Independence
P(A|B) = P(A)
Events are independent when knowing B happened gives no information about A. Otherwise they are dependent: knowledge of one updates the other.