Conditional Probability (Edexcel IAL Maths)

Set Notation

Spec point — 3.2

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

  • S, U, $ξ$ , $E$ 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“$notA$ ”, “$notB$”, “$notC$” are denoted by $A',B',C'$ etc (Strictly pronounced “A prime” but often called “A dash”) $A'$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 $A\cap B$

    • On a Venn diagram this would be the overlap between the bubble for event A and the bubble for event $B$
    • From Basic Probability, for independent events

$P(A\cap B)=P(A)\times P(B)$

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

  • 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

$P(A\cup B)=P(A)+P(B)$

• The other set you may come across in probability is the empty set The empty set has no elements and is denoted by $\emptyset$

• The intersection of mutually exclusive events is the empty set, $\emptyset$

• And finally, $P(A')=1-P(A)$

How do I find probabilities from sets?

• 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

• Do not try to do everything using a single diagram – whether given one in the question or using your own; use mini-Venn diagrams and shading 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

Spec point — 3.2

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 $\frac{5}{8}$.

• 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”

• 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

  • $\mathrm{P} ( A \cap B ) = \mathrm{P} ( A ) \times \mathrm{P} ( B \vert A )$
  • $\mathrm{P} ( A \cap B ) = \mathrm{P} ( B ) \times \mathrm{P} ( A \vert B )$
  • 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 Venn diagram below illustrates the probabilities of three events, $A,B\mathrm{and}C$.

(a) Find

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

(ii) $\mathrm{P} ( B \vert A ' )$

(iii) $\mathrm{P} ( C ' \vert A ' )$

(b) Show, in two different ways, that the events $B$and $C$ 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( | )

• If given a Venn diagram with all the separate probabilities you may find it easier to work out P(A), P(B) etc first

Two-Way Tables

Spec point — 3.2

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 ($\cap$ , 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 $P$ 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

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