Set Notation & Conditional Probability (Cambridge International A-Level Mathematics)

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 $ℕ$ , or the set of real numbers, denoted by $ℝ$

• In probability, set notation allows us to talk about the sample space and events within in it

  • $E$ , S, U and $ξ$ 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 $P(A')=1-P(A)$

  • AND is denoted by ∩ (intersection) OR is denoted by ∪ (union) (remember $A\cup B$ 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: $P(A\cap B)=P(A)\times P(B)$
  • For mutually exclusive events: $P(A\cup B)=P(A)+P(B)$

How do I solve problems given in set notation?

• Recognise the notation and symbols used and then interpret them in terms of AND ($\cap$), OR ($\cup$) 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. $(A\cup B)'=A'\cap B'$ $(A\cap B)'=A'\cup B'$
    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 $P(S)=2P(D)$, $P(S\cup D)=0.9$ and $P(S\cap D)=0.3$, find

(i) $P(S')$

(ii) $P(S'\cap D)$

(iii) $P(S\cup D')$

(iv) $P((S\cup D)')$

Answer:

Exam Hint

• 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 ($\cup$) or intersection ($\cap$) (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 $\frac{5}{8}$.

• 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 2 ^ nd is white 1 ^ st is white ) = 5 / 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

• The diagrams above also show two more conditional probability results

  • $\mathrm{P} ( A \cap B ) = \mathrm{P} ( A ) \times \mathrm{P} ( B \vert A )$

$\mathrm{P} ( A \cap B ) = \mathrm{P} ( B ) \times \mathrm{P} ( A \vert B )$

(These are essentially the same as letters are interchangeable)

• For independent events we know $P(A\cap B)=P(A)\times P(B)$ so

P ( B | A ) = (-- P ( A ) x P ( B ))/(-- P ( A ) ) = text P end text ( B )

and similarly

$\mathrm{P} ( A \vert B ) = \mathrm{P} ( A )$

• 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 $P(A)=0.4$ and $P(B)=0.5$. It is also known that $P(A\cap B)=0.2$ .

(a)

Find

(i) $P ( A \vert B )$

(ii) $P ( B \vert A ' )$

(iii) $P(A\cap B)(A\cup B)$

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

Answer:

Exam Hint

• There are now several symbols used from set notation in probability – make sure you are familiar with them

  • union ($\cup$)
  • intersection ($\cap$ )
  • 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

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 ($\cap$ , intersection), OR ($\cup$, 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

    • $P\cap Q$(intersection, AND) will be the cell where outcome meets outcome Q
    • $P\cup Q$(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

  • $\mathrm{P} ( A \cap B ) = \mathrm{P} ( A ) \times \mathrm{P} ( B \vert A )$
  • equation (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.

Answer:

Exam Hint

• 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