Two-step probability trees
State the draw rule first. R means red and B means blue. The worked rules below use three red and two blue counters, drawn uniformly without replacement
- Read branches from their node A second-stage branch gives the probability after the first result on that path. The two branches leaving one node add to 1. \(\displaystyle \frac12+\frac12=1\)
- Update after removal Without replacement, reduce the total by one and reduce the count of the colour just drawn. With replacement, the original counts are restored
- Multiply along one path Both stages must occur for that ordered path. Multiply the first probability by the second probability on the same path \(\displaystyle \Pr(RR)=\frac35\times\frac24=\frac3{10}\)
- Add disjoint paths Different complete paths cannot happen together in one trial. Add the probabilities of the paths belonging to the event \(\displaystyle \begin{gathered}\Pr(\text{same colour})\\=\Pr(RR)+\Pr(BB)\end{gathered}\)
- Check the whole tree All complete path probabilities add to 1. Colour paths can be unequal even when individual counters are selected uniformly
Worked example
A bag has three red and two blue counters. Draw two counters uniformly without replacement. Find the probability of different colours
- \(\displaystyle \Pr(RB)=\frac35\times\frac24=\frac3{10}\)After red, two blue counters remain among four
- \(\displaystyle \Pr(BR)=\frac25\times\frac34=\frac3{10}\)After blue, three red counters remain among four
- \(\displaystyle \Pr(E)=\frac3{10}+\frac3{10}=\frac35\)The two complete paths are disjoint, so add them
Common mistakes
Difficulty levels
Questions about this topic
What does a second-stage branch mean?+
It gives the probability of that result after the earlier result on its path
Why multiply along a path?+
The path requires the first result and then the second result. Use the second probability for that first result
Why add different paths?+
Distinct complete paths cannot both happen in one trial. Add the paths that satisfy the event
Does no replacement mean two reds are impossible?+
Only if the bag began with one red counter. With several red counters, a different red counter may be drawn second
Can a branch have probability zero?+
Yes. That next result is impossible at that node, and every path using that branch has probability zero
When can I count paths instead of weighting them?+
Only when every counted path is equally likely. Grouped colour paths usually have unequal probabilities
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 9: Probability and statistics