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 , or the set of real numbers, denoted by
 In probability, set notation allows us to talk about the sample space and events within in it
 , S, U and are common symbols used for the Universal set
In probability this is the entire sample space  Events are denoted by capital letters, A, B, C etc
 A' is called the complement of and means “not A”
(Strictly pronounced “ A prime” but often called “A dash”) Recall the important and easily missed result
 AND is denoted by ∩ (intersection)
OR is denoted by ∪ (union) (remember includes both)
 , S, U and are common symbols used for the Universal set
 The other set you may come across in probability is the empty set
The empty set has no elements and is denoted by ∅
The intersection of mutually exclusive events is the empty set,
 Set notation allows us to write probability results formally
 For independent events:
 For mutually exclusive events:
How do I solve problems given in set notation?
 Recognise the notation and symbols used and then interpret them in terms of AND (), OR () and/or NOT (‘) statements
 Venn diagrams lend themselves particularly well to deducing which sets or parts of sets are involved draw miniVenn 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.
Not convinced? Sketch a Venn diagram and shade it in!  In such questions it can be the unshaded part that represents the solution
 e.g.
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.
is the event a member selected the singles competition.
is the event a member selected the doubles competition.
Given that , and , find
(i)
(ii)
(iii)
(iv)
Exam Tip
 Venn diagrams are not expected but they are extremely useful
 Do not try to do everything on one diagram though  use miniVenn diagrams with shading (no values) for each part of a question
 Do double check whether you are dealing with union () or intersection () (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
Conditional Probability
What is conditional probability?
 You have already been using conditional probability in Tree Diagrams
 Probabilities change depending on the outcome of a prior event
 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 .
 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
 There are other phrases that imply or mean the same things as “given that”
 Tree diagrams are great for events that follow on from one another

 Otherwise Venn diagrams are extremely useful
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 experiment) has already occurred
 Otherwise Venn diagrams are extremely useful

 The diagrams above also show two more conditional probability results
(These are essentially the same as letters are interchangeable)
 For independent events we know so
and similarly
 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 probabilities of two events, and are described as and .
It is also known that .
Exam Tip
 There are now several symbols used from set notation in probability – make sure you are familiar with them
 union ()
 intersection ( )
 not (‘)
 given that (  )
 Use Venn diagrams to help deduce missing probabilities in questions – you may find it easier to work these out first before answering questions directly
TwoWay Tables
What are twoway tables?
 In probability, twoway 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
How do I solve problems given involving twoway tables?
 Questions will usually be wordy – and may not even mention twoway tables
 Questions will need to be interpreted in terms of AND ( , intersection), OR (, 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
 (intersection, AND) will be the cell where outcome meets outcome 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
 With event A outcomes on columns and event B outcomes on rows
(see Worked Example Q(b)(ii))
 You may need to use the results
 (for independent events)
Worked Example
The incomplete twoway 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 



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