Set Notation & Conditional Probability (CIE A Level Maths: Probability & Statistics 1)

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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 straight natural numbers  , or the set of real numbers, denoted by straight real numbers
  • In probability, set notation allows us to talk about the sample space and events within in it
    • calligraphic E , S, U and xi  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 bold P bold left parenthesis bold italic A bold apostrophe bold right parenthesis bold equals bold 1 bold minus bold P bold left parenthesis bold italic A bold right parenthesis
    • AND is denoted by ∩ (intersection)
      OR is denoted by ∪
      (union)     (remember A union B includes both)
  • 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:       straight P left parenthesis A intersection B right parenthesis equals straight P left parenthesis A right parenthesis cross times straight P left parenthesis B right parenthesis                           
    • For mutually exclusive events:  straight P left parenthesis A union B right parenthesis equals straight P left parenthesis A right parenthesis plus straight P left parenthesis B right parenthesis

How do I solve problems given in set notation?

  • Recognise the notation and symbols used and then interpret them in terms of AND (intersection), OR (union) 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

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

  • 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.   left parenthesis A union B right parenthesis apostrophe equals A apostrophe intersection B apostrophe
               left parenthesis A intersection B right parenthesis apostrophe equals A apostrophe union B apostrophe          
      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 straight P left parenthesis S right parenthesis equals 2 straight P left parenthesis D right parenthesis, P left parenthesis S union D right parenthesis equals 0.9 and P left parenthesis S intersection D right parenthesis equals 0.3, find

(i)   straight P left parenthesis S apostrophe right parenthesis

(ii)   straight P left parenthesis S apostrophe intersection D right parenthesis

(iii)   P left parenthesis S union D apostrophe right parenthesis

(iv)   P left parenthesis left parenthesis S union D right parenthesis apostrophe right parenthesis

 

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

  • Venn diagrams are not expected but they are extremely useful
    • Do not try to do everything on one diagram though - use mini-Venn diagrams with shading (no values) for each part of a question
  • Do double check whether you are dealing with union (begin mathsize 16px style union end style) or intersection (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

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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 begin mathsize 16px style 5 over 8 end style.

  • 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 text P( end text right enclose 2 to the power of nd space is space white end enclose space 1 to the power of st space is space white right parenthesis equals 5 over 8
    • 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

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

  • The diagrams above also show two more conditional probability results
    • straight P left parenthesis A intersection B right parenthesis equals straight P left parenthesis A right parenthesis cross times straight P left parenthesis B vertical line A right parenthesis

                    straight P left parenthesis A intersection B right parenthesis equals straight P left parenthesis B right parenthesis cross times straight P left parenthesis A vertical line B right parenthesis   

(These are essentially the same as letters are interchangeable)

  • For independent events we know straight P left parenthesis A intersection B right parenthesis equals straight P left parenthesis A right parenthesis cross times straight P left parenthesis B right parenthesis so

straight P left parenthesis B vertical line A right parenthesis equals fraction numerator horizontal strike straight P left parenthesis A right parenthesis end strike cross times straight P left parenthesis B right parenthesis over denominator horizontal strike straight P left parenthesis A right parenthesis end strike end fraction equals text P end text left parenthesis B right parenthesis

and similarly

straight P left parenthesis A vertical line B right parenthesis equals straight P left parenthesis A right parenthesis

  • 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, A and B are described as straight P left parenthesis A right parenthesis equals 0.4 and straight P left parenthesis B right parenthesis equals 0.5.
It is also known that P left parenthesis A intersection B right parenthesis equals 0.2 .

(a)
Find
(i)
P left parenthesis A vertical line B right parenthesis 
(ii)
P left parenthesis B vertical line A apostrophe right parenthesis
(iii)
P left parenthesis A intersection B right parenthesis left parenthesis A union B right parenthesis

(b)
Show, in two different ways, that the events A and B are independent.

2-3-1-cie-fig4-we2-solution

Examiner Tip

  • There are now several symbols used from set notation in probability – make sure you are familiar with them
    • union (begin mathsize 16px style union end style)
    • intersection (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

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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 (begin mathsize 16px style intersection end style , intersection), OR (begin mathsize 16px style union end style, 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
      • begin mathsize 16px style P intersection Q end style(intersection, AND) will be the cell where outcome  meets outcome Q
      • size 16px P size 16px union size 16px 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
    • begin mathsize 16px style straight P left parenthesis A intersection B right parenthesis equals straight P left parenthesis A right parenthesis cross times straight P left parenthesis B vertical line A right parenthesis end style
    • (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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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.