Two-step ordered sample spaces
State the experiment and replacement rule before listing outcomes or assigning probabilities
- Keep order The first entry records stage one. The second records stage two. Reversing unequal entries gives a different outcome \(\displaystyle (1,2)\ne(2,1)\)
- List systematically Fix a first result, list every possible second result, then move to the next first result. Exclude impossible pairs
- Replacement matters Let n be the number of distinct counters and N the total number of ordered outcomes. Returning a counter makes its label possible again. Keeping it out leaves n − 1 choices for the second draw \(\displaystyle \begin{gathered}N_{\text{return}}=n^2\\N_{\text{keep out}}=n(n-1)\end{gathered}\)
- Use equally likely outcomes For uniform individual selections or independent fair spinners, each legal ordered outcome has the same probability. Here n(E) counts favourable outcomes and N counts all possible outcomes. Divide the favourable count by the total \(\displaystyle \Pr(E)=\frac{n(E)}{N}\)
- Do not assume labels are equally likely Grouping several individual outcomes under one colour or sum can give groups of different sizes. Count the equally likely individual outcomes first
Worked example
A bag contains three counters labelled 1, 2 and 3. Draw two counters in order, selecting uniformly without replacement. Find the probability that at least one number is 1.
- \(\displaystyle N=3\times2=6\)Each first draw leaves two choices
- \(\displaystyle \begin{gathered}(1,2),(1,3),\\(2,1),(3,1)\end{gathered}\)These four ordered outcomes include label 1.
- \(\displaystyle \Pr(E)=\frac46=\frac23\)Divide favourable count by all six equally likely outcomes
Common mistakes
Difficulty levels
Questions about this topic
Why does order matter?+
The first and second entries record different stages. Reversing distinct results produces a different ordered outcome
What does replacement mean?+
Return the first selected counter and mix before selecting again. Its label can occur twice
Why is a diagonal cell impossible without replacement?+
Each label identifies one physical counter. Once it is removed, that same counter cannot be selected again
When can I divide a favourable count by a total count?+
When the listed individual outcomes are equally likely. Otherwise use their probabilities rather than treating every grouped label as equally likely
Can two different counters have the same colour without replacement?+
Yes. Removing one counter does not remove every counter of its colour. Distinguish individual labels from colours
Why does a grid help?+
Fixing one result per row and one per column makes missing, repeated and impossible ordered pairs easier to detect
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