Probability Venn Diagrams

Probability Venn Diagrams

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.

Examples

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)`