Set Notation & Conditional Probability (OCR A Level Maths A): Revision Note

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, ,  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 “ ”, “”, “” are denoted by  etc (Strictly pronounced “A  prime” but often called “A  dash”)  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 

    • On a Venn diagram this would be the overlap between the bubble for event A and the bubble for event 

    • From Basic Probability, for independent events

• The event A or B is called the union of events A and B , and the symbol  is used i.e.  A or B  is written as 

  • 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

• 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, 

• And finally,  

How do I find probabilities from sets?

• 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 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.                        Not convinced?  Sketch a Venn diagram and shade it in!

  • In such questions it can be the unshaded part that represents the solution

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 .

• 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”

• 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

• 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

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

How do I find probabilities from 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 ( , 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 

    • (union, OR) will be all the cells for outcomes  and  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

  • 

  • (for independent events)