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Set Notation & Conditional Probability (CIE A Level Maths: Probability & Statistics 1)
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 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 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
- 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)
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 () 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
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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 .
- 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 .
Examiner 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
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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
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
- 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 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 |
|
|
|
Find the probability
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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