GCSE(F),

Probability events can be allocated to sets. The sets can then be analysed by using **Set Theory** or **Venn Diagrams**.

The universal set consists of all possible events.

For a given outcome, all the events that are possible for that outcome will belong to the same set.

The probability of a given outcome is:

**P**(outcome) = `frac(text(number of members of set))(text(number of members in universal set))`

The probability of an event A not happening is P(not A). In set terms, this is shown as

**P**(A`),

and **P**(A) + **P**(A`) = 1.

When a probability can be an outcome in A *AND* B (A **intersection** B, or `A nn B`) then the overlap of sets A and B are considered.

When a probability is being considered where an event can be in either outcome A *OR* B then all the elements in A and B are considered (elements in A **union** B, or `A uu B`: note that elements in both A and B are counted only once.

1. Two fair dice are thrown. Using a Venn Diagram, what is the probability of `O uu P` where O is the set of odd numbers and P is the set of Prime numbers?

Answer: `frac(6)(11)`

`epsilon = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}`

`text(O) = {3, 5, 7, 9, 11}`

`text(P) = {2, 3, 5, 7, 11}`

`O uu P = {2, 3, 5, 7, 9, 11}` (events in either set)

Number of events = 11

**P**(`O uu P`) = `frac(6)(11)`

2. With the Venn Diagram, above, what is the probability of `O nn P`?

Answer: `frac(4)(11)`

`O nn P = {3, 5, 7, 11}` = 4 events

`epsilon` is 11 events

**P**`(O nn P) = frac(4)(11)`

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