# 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:

## Probability Formulae

**Addition Formula**:

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.**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.