Conditional Probability and Tree Diagrams

Conditioning shrinks the sample space

The conditional probability of $A$ given $B$ is the chance of $A$ once you know $B$ has happened:

Rearranged, this is the general multiplication rule $P(A\cap B)=P(B)\,P(A\mid B)$, which works whether or not the events are independent. From a two-way table, $P(A\mid B)$ is the count in the $A$-and-$B$ cell divided by the row or column total for $B$ — not by the grand total. Independence is the special case $P(A\mid B)=P(A)$: knowing $B$ changes nothing.

Tree diagrams

A tree diagram organises a two-stage process: first-stage branches carry unconditional probabilities, second-stage branches carry conditional ones. Multiply along a path to get the joint probability of that path; add across paths that lead to the same outcome. Example: a disease affects $2\%$ of people; the test is positive for $95\%$ of the sick and for $8\%$ of the healthy. Then $P(\text{positive})=0{,}02\cdot0{,}95+0{,}98\cdot0{,}08=0{,}019+0{,}0784=0{,}0974$.

Reversing the condition

The exam's favourite question runs the tree backwards: given a positive test, what is the chance of disease?

Only about one positive in five is a true case, because the healthy group is so much larger that its $8\%$ false positives outnumber the sick group's true positives. The numerator is one path; the denominator is every path ending in "positive".