2.3.1 Set Notation & Conditional Probability

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)
• 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 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

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

• 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

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 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 ( , 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

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

• You may need to use the results
  • 
  • (for independent events)