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Conditional Probability

▶︎  Watch it animatedinteractive step-through · ~3 min · optional ⚙︎  Open the appletConditional Lab · one shaded block, two denominators, two answers nowhere near each other

Conditional probability restricts the sample space: $P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}$ counts the outcomes in both events against the size of the condition, which in a two-way table is a single row or column. Rearranged, it gives the multiplication rule $P(A \text{ and } B) = P(B) \cdot P(A \mid B)$, which is what a tree diagram encodes: multiply along a branch, add across the paths that produce the event. Within a fixed condition the conditional distribution still sums to 1, so $P(A \mid B) + P(A^c \mid B) = 1$.

The direction of the conditional is the error that costs the most. $P(A \mid B)$ and $P(B \mid A)$ share the joint numerator and divide by different totals, so a test that returns positive for 95% of people with a disease can still be wrong about most positives: with 1% prevalence, 95 true positives arrive alongside 990 false ones and $P(\text{disease} \mid \text{positive}) = \frac{95}{1085} \approx 0.088$. The related failure is dividing by the grand total instead of the condition, which converts a conditional into a joint probability and quietly changes the population the claim is about.

10,000 people. prevalence 1%, sensitivity 95%, specificity 90% 0.01 0.99 disease 100 healthy 9900 0.95 0.05 0.10 0.90 positive 95 negative 5 positive 990 negative 8910 P(disease | positive) = 95 / 1085 = 0.088 1085 positives in all
Multiply along a branch, add across the paths. The healthy group is 99 times larger, so its 10% error rate contributes over ten times more positives than the disease branch does.
the same 95, two different worlds positive negative total disease 95 5 100 healthy 990 8910 9900 total 1085 8915 10000 row: P(positive | disease) = 95 / 100 = 0.95 column: P(disease | positive) = 95 / 1085 = 0.088 joint: 95 / 10000 = 0.0095, a third question again
A row and a column intersect in one cell and divide by different totals. Which line you read across is decided by the word given, not by which event sounds more important.

The work

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Lesson
Conditional Probability ›

Builds conditional probability as a restricted sample space, reads it off rows and columns of a table, runs the multiplication rule through tree diagrams, and works the base-rate case where a 95% accurate test is right about 9% of its positives.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items on conditioning: the reversed conditional, the grand total slipped under a conditional, tree branches multiplied in the wrong order, and base rates ignored. Take it cold to find your habit, or after the lesson to check it is gone.

Not started · 10 items · ~15 min
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Targeted Practice
Drill a single misconception ›

Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.

Take the diagnostic to identify your misconceptions
Diagnostic first