Further Tree Diagrams (CIE A Level Maths: Probability & Statistics 1)

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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
    • questions may use set notation as well, or alongside contextual questions begin mathsize 16px style union end style(union),  intersection(intersection) , ‘ (complement), | ("given that")
    • more detailed use of conditional probability
    • three events for each experiment and three experiments could be used

2-3-2-cie-fig0-three-tree

How do I solve conditional probability problems using tree diagrams?

  • Interpreting questions in terms of AND (intersection), OR (begin mathsize 16px style union end style), complement ( ‘ ) and “given that” ( | )
  • Condition probability may now be involved too
  • 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, begin mathsize 16px style P left parenthesis B vertical line A right parenthesis end style then event A has already occurred so start by looking for the branch event A in the 1st experiment, and then P(B | A)  would be the branch for event B in the 2nd experiment

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

3-2-3-fig1-tree-setup

  • The diagram above gives rise to some probability formulae you will see in Probability Formulae
  • Error converting from MathML to accessible text. (“given that”) is the probability on the branch of the 2nd experiment
  • However, the “given that” statement Error converting from MathML to accessible text. 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 P left parenthesis F apostrophe intersection W right parenthesis equals 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 W didn’t occur.

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

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

Examiner Tip

  • Be wary of assuming that “given that” statements will always be referring to something on the second set of branches (2nd experiment), they can work the other way!

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Paul

Author: Paul

Expertise: Maths

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.