Compound Probability

Tree Diagram

Represent sequential outcomes as nested branches and multiply branch probabilities to get leaf outcome probabilities.

Leaf probabilities can be summed to verify they total 1.

Example

Use a path table to multiply branch probabilities and add outcomes.

highlighted = computed this step

Step 1 — Branch probabilities

List the branch probability for each split.

branches=2p=12branches= \hl{2} \quad p= \hlmath{\frac{1}{2}}

Step 2 — Path HH

Multiply branch probabilities for path HH.

pathfirstpasecondpbprodHHH12H1214HT□□□□□TH□□□□□TT□□□□□\begin{array}{cccccc}\text{path} & \text{first} & \text{pa} & \text{second} & \text{pb} & \text{prod} \\ \text{HH} & \text{H} & \hlmath{\frac{1}{2}} & \text{H} & \hlmath{\frac{1}{2}} & \hlmath{\frac{1}{4}} \\ \text{HT} & \square & \square & \square & \square & \square \\ \text{TH} & \square & \square & \square & \square & \square \\ \text{TT} & \square & \square & \square & \square & \square\end{array}

Step 3 — Path HT

Multiply branch probabilities for path HT.

pathfirstpasecondpbprodHHH12H1214HTH12T1214TH□□□□□TT□□□□□\begin{array}{cccccc}\text{path} & \text{first} & \text{pa} & \text{second} & \text{pb} & \text{prod} \\ \text{HH} & \text{H} & \frac{1}{2} & \text{H} & \frac{1}{2} & \frac{1}{4} \\ \text{HT} & \text{H} & \hlmath{\frac{1}{2}} & \text{T} & \hlmath{\frac{1}{2}} & \hlmath{\frac{1}{4}} \\ \text{TH} & \square & \square & \square & \square & \square \\ \text{TT} & \square & \square & \square & \square & \square\end{array}

Step 4 — Path TH

Multiply branch probabilities for path TH.

pathfirstpasecondpbprodHHH12H1214HTH12T1214THT12H1214TT□□□□□\begin{array}{cccccc}\text{path} & \text{first} & \text{pa} & \text{second} & \text{pb} & \text{prod} \\ \text{HH} & \text{H} & \frac{1}{2} & \text{H} & \frac{1}{2} & \frac{1}{4} \\ \text{HT} & \text{H} & \frac{1}{2} & \text{T} & \frac{1}{2} & \frac{1}{4} \\ \text{TH} & \text{T} & \hlmath{\frac{1}{2}} & \text{H} & \hlmath{\frac{1}{2}} & \hlmath{\frac{1}{4}} \\ \text{TT} & \square & \square & \square & \square & \square\end{array}

Step 5 — Path TT

Multiply branch probabilities for path TT.

pathfirstpasecondpbprodHHH12H1214HTH12T1214THT12H1214TTT12T1214\begin{array}{cccccc}\text{path} & \text{first} & \text{pa} & \text{second} & \text{pb} & \text{prod} \\ \text{HH} & \text{H} & \frac{1}{2} & \text{H} & \frac{1}{2} & \frac{1}{4} \\ \text{HT} & \text{H} & \frac{1}{2} & \text{T} & \frac{1}{2} & \frac{1}{4} \\ \text{TH} & \text{T} & \frac{1}{2} & \text{H} & \frac{1}{2} & \frac{1}{4} \\ \text{TT} & \text{T} & \hlmath{\frac{1}{2}} & \text{T} & \hlmath{\frac{1}{2}} & \hlmath{\frac{1}{4}}\end{array}

Step 6 — Sum paths

Add all path probabilities.

14+14+14+14=1\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \hl{1}
tree-diagram Product of branch probabilities gives leaf probabilities.