Tim Gan Math
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Tree Diagrams

Tree Diagrams

Tree diagrams

Tree diagrams are useful for a sequence of events, particularly with conditional probability or infinitely many possible full outcomes.

  1. First-level branches represent the probabilities of the initial events.
  2. Second-level branches represent conditional probabilities.
  3. Each complete root-to-leaf path gives an intersection probability.
Two-stage probability tree
\(a\)
\(A\)
\(1-a\)
\(A'\)
\(b\)
\(B\)
\(1-b\)
\(B'\)
\(c\)
\(B\)
\(1-c\)
\(B'\)
Probabilities read from the tree

\[P(A)=a,\quad P(A')=1-a\]\[P(B)=ab+(1-a)c,\quad P(B')=a(1-b)+(1-a)(1-c)\]\[P(A\cap B)=ab,\quad P(A\cap B')=a(1-b)\]\[P(A'\cap B)=(1-a)c,\quad P(A'\cap B')=(1-a)(1-c)\]\[P(B\mid A)=b,\quad P(B'\mid A)=1-b,\quad P(B\mid A')=c,\quad P(B'\mid A')=1-c\]

Independence

Assume \(0<a<1\), so both conditional branches are defined. Then \(A\) and \(B\) are independent if and only if \(b=c\).

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