Source-first probability trees
Separate choosing a source from choosing an outcome within that source
- Separate source and result A first-stage probability chooses a source. A conditional probability describes what can happen after that source
- Multiply one path Both stages occur, so multiply the source probability by the conditional result probability \(\displaystyle \Pr(A\cap R)=\Pr(A)\Pr(R\mid A)\)
- Add disjoint paths The source choices cannot occur together. Add the relevant path probabilities to find the total event probability \(\displaystyle \Pr(R)=\Pr(A\cap R)+\Pr(B\cap R)\)
- Use the right denominator For a conditional branch use the total in its source, not the combined contents of every source
- Read event language Exactly one excludes the case where both conditions hold. At least one includes it
- Check the experiment Counts imply probabilities only when the stated selection rule makes the individual items equally likely
Worked example
A is chosen with probability 1/4 and B with probability 3/4. Red has conditional probability 3/4 after A and 1/4 after B. Find Pr(red)
\[\begin{aligned}&\Pr(R)\\&=\Pr(A\cap R)+\Pr(B\cap R)\end{aligned}\]
- \(\displaystyle \begin{aligned}&\Pr(A\cap R)\\&=\frac14\times\frac34\\&=\frac3{16}\end{aligned}\)Weight the red result from A by the chance of selecting A.
- \(\displaystyle \begin{aligned}&\Pr(B\cap R)\\&=\frac34\times\frac14\\&=\frac3{16}\end{aligned}\)Use the conditional row belonging to B.
- \(\displaystyle \begin{aligned}&\Pr(R)\\&=\frac3{16}+\frac3{16}\\&=\frac38\end{aligned}\)The two source paths are disjoint
Common mistakes
Unweighted average
✕\(\displaystyle \begin{aligned}&\Pr(R)\\&=(\frac34+\frac14)/2\\&=\frac12\end{aligned}\)✓\(\displaystyle \begin{aligned}&\Pr(R)\\&=\frac14\times\frac34+\frac34\times\frac14\\&=\frac38\end{aligned}\)
Source A and B have different selection probabilitiesConditional versus joint
✕\(\displaystyle \begin{aligned}&\Pr(A\cap R)\\&=\frac34\end{aligned}\)✓\(\displaystyle \begin{aligned}&\Pr(A\cap R)\\&=\frac14\times\frac34\\&=\frac3{16}\end{aligned}\)
The joint event includes selecting A.Exactly one versus at least one
✕\(\displaystyle \begin{aligned}&\Pr(\text{exactly one})\\&=\frac7{16}\end{aligned}\)✓\(\displaystyle \begin{aligned}&\Pr(\text{exactly one})\\&=\frac1{16}+\frac3{16}\\&=\frac14\end{aligned}\)
Exclude A then red because both conditions holdDifficulty levels
introUse two or three sources for a path, total or at-least-one event
standardCombine source weights and outcome probabilities, including exactly-one events
challengeInterpret the full event range and calculate six missing probabilities
examOrganise exact probabilities across multiple sources and favourable outcomes
Questions about this topic
Why multiply along a path?+
Both stages must occur. Multiply the first probability by the second probability conditional on the first
Why add different paths?+
Complete ordered paths are disjoint alternatives
Can I average source probabilities?+
Only an equally weighted average is appropriate when sources are equally likely. Otherwise use their selection weights
Do unequal source sizes determine selection weights?+
No. Read the stated source-selection mechanism; it can use separate tickets or assigned probabilities
What does a question mark mean?+
Calculate that branch probability from the supplied counts and the relevant denominator
What does exactly one mean?+
One condition holds and the other does not. Exclude cases where both hold
Must siblings sum to one?+
Yes, when they list every possible next outcome after the same history
Does multiplication require independence?+
No. Multiplication along a tree uses conditional branches, which may change after earlier outcomes
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 4: Probability
In this chapter
Describe and order probability valuesSingle-step theoretical probability and complements4A Probability scale and experiments: Experimental probability from observed frequenciesFinite events, probability assignments and complements4A Probability scale and experiments: Design and interpret a probability simulation4B Venn diagrams and sets: Venn regions, set language and probabilityCompound events, the addition rule and mutual exclusivityComplete and use a two-way probability tableConstruct and complete two-way frequency tablesConvert between Venn diagrams and two-way tablesConditional probability from exact representations4D Conditional probability: Conditional probability on Venn diagramsTwo-step sample-space tables with and without replacementRepresent a multi-stage finite sample spaceTree-diagram probability for two-stage experimentsMulti-stage probability, multiplication and total probability4G Independence and simulations: Independence on Venn diagramsTest exact independence4G Independence and simulations: Design and interpret a probability simulation4G Independence and simulations: Experimental probability from observed frequencies