Set Notation & Conditional Probability (AQA A Level Maths: Statistics)

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Paul

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17 July 2023

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Set Notation

What is set notation?

Set notation is a formal way of writing groups of numbers (or other mathematical entities such as shapes) that share a common feature – each number in a set is called an element of the set
- You should have come across common sets of numbers such as the natural numbers, denoted by $ℕ$ , or the set of real numbers, denoted by $ℝ$
In probability, set notation allows us to talk about the sample space and events within in it
- S, U, $ξ$ , $E$ are common symbols used for the Universal set
  In probability this is the entire sample space, or the rectangle in a Venn diagram
- Events are denoted by capital letters, A, B, C etc
- The events “ $n o t A$ ”, “ $n o t B$ ”, “ $n o t C$ ” are denoted by $A', B', C'$ etc
  (Strictly pronounced “A prime” but often called “A dash”)
  $A'$ is called the complement of A
In probability we are often looking at combined events
- The event A and B is called the intersection of events A and B , and the symbol ∩ is used
  i.e. A and B is written as $A \cap B$
  - On a Venn diagram this would be the overlap between the bubble for event A and the bubble for event $B$
  - From Basic Probability, for independent events

$P (A \cap B) = P (A) \times P (B)$

- The event A or B is called the union of events A and B , and the symbol $\cup$ is used
  i.e. A or B is written as $A \cup B$
  - On a Venn diagram this would be both the bubbles for event A and event B including their overlap (intersection)
  - From Basic Probability, for mutually exclusive events

$P (A \cup B) = P (A) + P (B)$

- The other set you may come across in probability is the empty set
  The empty set has no elements and is denoted by $\emptyset$

The intersection of mutually exclusive events is the empty set, $\emptyset$

And finally, $P (A') = 1 - P (A)$

3-2-1-fig1-venn-and-set-notation

How do I solve problems given in set notation?

Recognise the notation and symbols used and then interpret them in terms of AND ( $\cap$ ), OR ( $\cup$ ) and/or NOT (‘) statements
Venn diagrams lend themselves particularly well to deducing which sets or parts of sets are involved- draw mini-Venn diagrams and shade them
Practice shading various parts of Venn diagrams and then writing what you have shaded in set notation
With combinations of union, intersection and complement there may be more than one way to write the set required
- e.g. $(A \cup B)' = A' \cap B'$
  $(A \cap B)' = A' \cup B'$
  Not convinced? Sketch a Venn diagram and shade it in!
- In such questions it can be the unshaded part that represents the solution

Worked example

The members of a local tennis club can decide whether to play in a singles competition, a doubles competition, both or neither.

Once all members have made their choice the chairman of the club selects, at random, one member to interview about their decision.

$S$ is the event a member selected the singles competition.

$D$ is the event a member selected the doubles competition.

Given that $P (S) = 2 P (D), P (S \cup D) = 0.9$ and $P (S \cap D) = 0.3$ , find

(i) $P (S')$

(ii) $P (S' \cap D)$

(iii) $P (S \cup D')$

(iv) $P ((S \cup D)')$

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

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

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

Examiner Tip

Do not try to do everything using a single diagram – whether given one in the question or using your own; use mini-Venn diagrams and shading for each part of a question
Do double check whether you are dealing with union ( $\cup$ ) or intersection ( $\cap$ ) (or both) – when these symbols are used several times near each other in a question, it is easy to get them muddled up or misread them

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Conditional Probability

What is conditional probability?

Conditional probability is where the probability of an event happening can vary depending on the outcome of a prior event

You have already been using conditional probability
e.g. drawing more than one counter/bead/etc from a bag without replacement

(Note that, mathematically, that drawing one, not replacing then

drawing another is the same as drawing two at the same time.)

Consider the following example

e.g. Bag with 6 white and 3 red buttons. One is drawn at random and not replaced. A second button is drawn. The probability that the second button is white given that the first button is white is $\frac{5}{8}$ .

The key phrase here is “given that” – it essentially means something has already happened.
- In set notation, “given that” is indicated by a vertical line ( | ) so the above example would be written $P(2^{nd} is white 1^{st} is white) = \frac{5}{8}$
- There are other phrases that imply or mean the same things as “given that”
Venn diagrams are helpful again but beware – the denominator of fractional probabilities will no longer be the total of all the frequencies or probabilities shown
- “given that” questions usually reduce the sample space as an event (a subset of the outcomes of the first event) has already occurred

--tR8mHB_3-2-1-fig3-cp-venn

The diagrams above also show two more conditional probability results
- $P (A \cap B) = P (A) \times P (B | A)$

$P (A \cap B) = P (B) \times P (A | B)$

(These are essentially the same as letters are interchangeable)

For independent events we know $P (A \cap B) = P (A) \times P (B)$ so

$P (B | A) = \frac{P (A) \times P (B)}{P (A)} = P (B)$

and similarly

$P (A | B) = P (A)$

The independent result should make sense logically – if events A and B are independent then the fact that event B has already occurred has no effect on the probability of event A happening

Worked example

The Venn diagram below illustrates the probabilities of three events, $A, B and C$ .

3-2-1-fig4-we2-diagram

(a)

Find

(i)

P (A | B)

(ii)

P (B | A')

(iii)

P (C' | A')

(b)

Show, in two different ways, that the events

B

and

C

are independent.

(a)

Find

(i)

P (A | B)

(ii)

P (B | A')

(iii)

P (C' | A')

3-2-1-fig4-we2-solution-part-1

3-2-1-fig4-we2-solution-part-2

(b)

Show, in two different ways, that the events

B

and

C

are independent.

Examiner Tip

There are now several symbols used from set notation in probability – make sure you are familiar with them
- union ( $\cup$ )
- intersection ( $\cap$ )
- not (‘)
- given that ( | )
If given a Venn diagram with all the separate probabilities you may find it easier to work out P(A), P(B) etc first

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Two-Way Tables

What are two-way tables?

In probability, two-way tables list the frequencies for the outcomes of two events – one event along the top (columns), one event down the side (rows)
The frequencies, along with a “Total” row and “Total” column instantly show the values involved in finding probabilities

3-2-1-fig5-two-way-and-notation

How do I solve problems given involving two-way tables?

Questions will usually be wordy – and may not even mention two-way tables
- Questions will need to be interpreted in terms of AND ( $\cap$ , intersection), OR ( $\cup$ , union), NOT (‘) and GIVEN THAT ( | )
Complete as much of the table as possible from the information given in the question
- If any empty cells remain, see if they can be calculated by looking for a row or column with just one missing value
Each cell in the table is similar to a region in a Venn diagram
- With event A outcomes on columns and event B outcomes on rows
  - $P \cap Q$ (intersection, AND) will be the cell where outcome $P$ meets outcome $Q$
  - $P \cup Q$ (union, OR) will be all the cells for outcomes $P$ and $Q$ including the cell for both
- Beware! As union includes the cell for both outcomes, avoid counting this cell twice when calculating frequencies or probabilities

(see Worked Example Q(b)(ii))

You may need to use the results
- $P (A \cap B) = P (A) \times P (B | A)$
- $P (A | B) = P (A)$ (for independent events)

Worked example

The incomplete two-way table below shows the type of main meal provided by 80 owners to their cat(s) or dog(s).

	Dry Food	Wet Food	Raw Food	Total
Dog	11		8
Cat		19		33
Total	21

(a)

Complete the two-way table

(b)

One of the 80 owners is selected at random.
Find the probability

(i)

the selected owner has a cat and feeds it raw food for its main meal.

(ii)

the selected owner has a dog or feeds it wet food for its main meal.

(iii)

the owner feeds raw food to its pet, given it is a dog.

(iv)

the owner has a cat, given that they feed it dry food.

3-2-1-fig6-we3-solution

Examiner Tip

Ensure any table – given or drawn - has a “Total” row and a “Total” column
Do not confuse a two-way table with a sample space diagram – a two-way table does not necessarily display all outcomes from an experiment, just those (events) we are interested in

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Previous:3.1.3 Tree DiagramsNext:3.2.2 Further Venn Diagrams

Set Notation & Conditional Probability (AQA A Level Maths: Statistics)

Revision Note

What is set notation?

How do I solve problems given in set notation?

What is conditional probability?

What are two-way tables?

How do I solve problems given involving two-way tables?

You've read 0 of your 5 free revision notes this week

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Join the 100,000+ Students that ❤️ Save My Exams

Author: Paul