# More probability

We learned about a number of probability topics in AS Maths including Venn Diagrams, Tree Diagrams and the definitions of Mutual Exclusivity and Independence. In A2 Maths, we take a closer look at conditional probabilities and mutual exclusivity & independence with a particular focus on set notation and introduce some useful formulae.

## Conditional Probability

Conditional probabilities are probabilities that change given the occurrence of a previous event or given additional information about an outcome. We have seen conditional probabilities in tree diagrams:

The tree diagram above shows the different outcomes when drawing two counters without replacement from a bag containing 4 red and 6 blue counters – see more Tree Diagram examples. The probabilities change depending on the outcome of the first counter drawn – this is an example of conditional probability. Conditional probabilities may also be presented in a two-way table (or contingency table) – see Example 1. We represent a conditional probability as and we read it as ‘the probability that A occurs given that B has occurred’ or ‘the probability of A given information of B’. A conditional probability question for a tree diagram can also be seen in Example 1.

## Set Notation

We can also calculate conditional probabilities using Venn diagrams by using prior information of an outcome to restrict the sample space. Before we do so, we take a brief look at set notation. We saw the following definitions when learning about Venn diagrams in AS Maths. We now introduce the corresponding set notation:

or Â

Â

## Probability Formulae

If you colour in event A in one colour, then event B in another, you will notice that the intersection was coloured in twice. Hence, in order to find the probability of the union, one must add the respective probabilities of events A and B whilst removing a single probability corresponding to one of the intersections.
2. Multiplication Formula:

This can be seen in a Venn diagram – given the information that B has occurred, the new full sample space becomes the set B. The probability that A has occurred is then the part of A that lies in the intersection. It can be thought of as the chances of being in A given that you are in B is the probability of the intersection out of the probability of B. It follows from the formula that . This can be seen in a tree diagram – given that B has occurred follow event B along the first branch with probability . The probability that A then occurs is following event A along the second branch. Alternatively, , which can be seen from the original formula by swapping A and B around.

## Mutually Exclusive and Independent Events

We saw the definitions of mutual exclusivity and independence in AS Maths. If events are mutually exclusive, they do not intersect on a Venn diagram. It follows that…

for mutually exclusive events.

For independent events, – the fact that event B occurred does not affect the probability of A occurring. From the multiplication formation it follows that and so…

for independent events.

See Example 2 for a demonstration of how to use the formulae and how to determine if events are mutually exclusive or independent.

## Examples

The following two-way table shows the eye and hair colour of 100 university maths students:

Let be the event that a student has blue eyes, be the event that a student has brown eyes, if they have dark hair and if they have light hair.

a) For a student drawn at random, find the following probabilities:
i)
ii)
iii)
iv)

b) Show this information in a tree diagram.

c) Use the tree diagram to find and .

a)
i) According to the table, 24 out of 100 students have blue eyes and so .
ii) Similarly,
iii) Of the 24 people that have blue eyes, 10 have dark hair and so the probability of dark hair given blue eyes is , that is, .
iii) Of the 76 people that have brown eyes, 22 have light hair and so the probability of light hair given brown eyes is , that is, .
b)

c) By following the branches through the tree diagram, blue eyes then light hair, we see that . Similarly, not blue eyes (or brown eyes) then light hair, gives .

Given that , and ,

a) Find .
b) Determine whether the events and are mutually exclusive and/or independent or not.
c) Find .
d) Show the information in a Venn diagram and determine
i) and
ii)

a) We use the addition formula: or and rearrange to get .
b) Since , events and are not mutually exclusive. Events are independent if – in this case, whereas and so the events aren’t independent either.
c) Using the multiplication formula: , it follows that .
d)

To find we restrict the sample space to not B and look at the part of A that is not in B. This is 0.4 and since it is out of 0.75, we have .

To find we restrict the sample space to and look at the part of that lies in it. All of is in and so .