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 $\cup$ (union), $\cap$ (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 ( $\cap$ ), OR ( $\cup$ ) 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, $P (B | A)$ then event A has already occurred so start by looking for the branch event A in the 1^st experiment, and then would be the branch for event in the 2^nd experiment

Similarly, $P (B | A)$ would require starting with event “ $n o t A$ ” in the 1^st experiment and event B in the 2^nd experiment

The diagram above gives rise to some probability formulae you will see in the next revision note
$P (B | A)$ (“given that”) is the probability on the branch of the 2^nd experiment
However, the “given that” statement $P (A | B)$ is more complicated and a matter of working backwards
- from Conditional Probability, $P (A | B) = \frac{P (A \cap B)}{P (B)}$
- from the diagram above, $P (B) = P (A \cap B) + P (A' \cap B)$
- leading to $P (A | B) = \frac{P (A \cap B)}{P (A \cap B) + P (A' \cap B)}$
- 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 $P (F' \cap W) = 0.15$ .

Find

(i)

P (W | F')

(ii)

P (F | W')

(iii)

the probability that event

F

didn’t occur, given that event

W

didn’t occur.

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

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

Examiner 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 $P (not A) = 1 - P (A)$
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

Further Tree Diagrams (AQA A Level Maths: Statistics)

Revision Note

What do you mean by further tree diagrams?

How do I solve conditional probability problems using tree diagrams?

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1. Statistical Sampling

1.1 Sampling & Data Collection

1.1.1 Sampling & Data Collection

2. Data Presentation & Interpretation

2.1 Statistical Measures

2.1.1 Basic Statistical Measures

2.1.2 Frequency Tables

2.1.3 Standard Deviation & Variance

2.2 Data Presentation

2.2.1 Data Presentation

2.2.2 Box Plots & Cumulative Frequency

2.2.3 Histograms

2.3 Working with Data

2.3.1 Outliers & Cleaning Data

2.3.2 Intrepreting Data

2.3.3 Skewness

2.4 Correlation & Regression

2.4.1 Correlation & Regression

2.5 Further Correlation & Regression (A Level only)

2.5.1 PMCC & Non-linear Regression

2.5.2 Hypothesis Testing for Correlation

3. Probability

3.1 Basic Probability

3.1.1 Calculating Probabilities & Events

3.1.2 Venn Diagrams

3.1.3 Tree Diagrams

3.2 Further Probability (A Level only)

3.2.1 Set Notation & Conditional Probability

3.2.2 Further Venn Diagrams

3.2.3 Further Tree Diagrams

3.2.4 Probability Formulae

4. Statistical Distributions

4.1 Probability Distributions

4.1.1 Discrete Probability Distributions

4.2 Binomial Distribution

4.2.1 The Binomial Distribution

4.2.2 Calculating Binomial Probabilities

4.3 Normal Distribution (A Level only)

4.3.1 The Normal Distribution

4.3.2 Normal Distribution - Calculations

4.3.3 Standard Normal Distribution

4.4 Choosing Distributions (A Level only)

4.4.1 Modelling with Distributions

4.4.2 Normal Approximation of Binomial

5. Hypothesis Testing

5.1 Hypothesis Testing

5.1.1 Hypothesis Testing

5.2 Hypothesis Testing (Binomial Distribution)

5.2.1 Binomial Hypothesis Testing

5.3 Hypothesis Testing (Normal Distribution) (A Level only)

5.3.1 Sample Mean Distribution

5.3.2 Normal Hypothesis Testing

6. Large Data Set

6.1 Large Data Set

6.1 Large Data Set

Author: Paul