AQA A Level Maths: Statistics

Revision Notes

3.2.3 Further Tree Diagrams

Test Yourself

Further Tree Diagrams

What do you mean by further tree diagrams?

  • The tree diagrams used here are no more complicated than those in the first Tree Diagrams revision note, however
    • The wording/terminology used in questions and on diagrams may now involve the use of set notation including the symbols begin mathsize 16px style union end style(union),  begin mathsize 16px style intersection end style(intersection) and ‘ (complement)
      • e.g.  P(A') would be used for “P(not A)”
    • Conditional probability questions can be solved using tree diagrams

How do I solve conditional probability problems using tree diagrams?

  • Interpreting questions in terms of AND (begin mathsize 16px style intersection end style), OR (union) and complement ( ‘ )
  • Condition probability may now be involved too - “given that” ( | )
  • This makes it harder to know where to start and how to complete the probabilities on a tree diagram
    • e.g. If given, possibly in words, P left parenthesis B vertical line A right parenthesis then event A has already occurred so start by looking for the branch event A in the 1st experiment, and then  would be the branch for event  in the 2nd experiment

Similarly, begin mathsize 16px style straight P left parenthesis B vertical line A right parenthesis end style would require starting with event “bold italic n bold italic o bold italic t bold space bold italic A  in the 1st experiment and event B in the 2nd experiment

UclzomJM_3-2-3-fig1-tree-setup

 

  • The diagram above gives rise to some probability formulae you will see in the next revision note
  • bold P bold left parenthesis bold italic B bold vertical line bold italic A bold right parenthesis (“given that”) is the probability on the branch of the 2nd experiment
  • However, the “given that” statement bold P bold left parenthesis bold italic A bold vertical line bold italic B bold right parenthesis is more complicated and a matter of working backwards
    • from Conditional Probability,  straight P left parenthesis A vertical line B right parenthesis equals fraction numerator straight P left parenthesis A intersection B right parenthesis over denominator straight P left parenthesis B right parenthesis end fraction
    • from the diagram above, P left parenthesis B right parenthesis equals P left parenthesis A intersection B right parenthesis plus P left parenthesis A apostrophe intersection B right parenthesis
    • leading to  bold P bold left parenthesis bold italic A bold vertical line bold italic B bold right parenthesis bold equals fraction numerator bold P bold left parenthesis bold A bold intersection bold B bold right parenthesis over denominator bold P bold left parenthesis bold A bold intersection bold B bold right parenthesis bold plus bold P bold left parenthesis bold A bold apostrophe bold intersection bold B bold right parenthesis end fraction
    • This is quite a complicated looking formula to try to remember so use the logical steps instead – and a clearly labelled tree diagram!

Worked example

The event F has a 75% probability of occurring.

The event W follows event F, and if event F has occurred, event W has an 80% chance of occurring.

It is also known that straight P left parenthesis F apostrophe intersection W right parenthesis space equals space 0.15 .

Find

(i)
straight P left parenthesis W vertical line F apostrophe right parenthesis
(ii)
straight P left parenthesis F vertical line W apostrophe right parenthesis
(iii)
the probability that event F didn’t occur, given that event Wdidn’t occur.

 

3-2-3-fig2-we-solution-part-1

2ACeFam__3-2-3-fig2-we-solution-part-2

Exam Tip

  • It can be tricky to get a tree diagram looking neat and clear first attempt – it can be worth drawing a rough one first, especially if there are more than two outcomes or more than two events; do keep an eye on the exam clock though!
  • Always worth another mention – tree diagrams make particularly frequent use of the result begin mathsize 16px style straight P left parenthesis not space A right parenthesis equals 1 minus straight P left parenthesis A right parenthesis end style
  • Tree diagrams have built-in checks
    • the probabilities for each pair of branches should add up to 1
    • the probabilities for each outcome of combined events should add up to 1

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Author: Paul

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.