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Chapter Overview

Learning Goals

**Learning Goals**

- Use Venn diagrams to display sets of data
- Use Venn diagrams to find probabilities

Maths

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A set is a collection of 'things' known as elements. In statistics, sets are usually presented on a Venn diagram. This provides an easy way for the probability of an element, inside a given set, to be worked out.

Sets are represented by a list of elements inside two curly brackets: { }

*The set of even numbers between *$1$* and *$10$*:*

*{*$2,4,6,8$*}**{*$x:$*is an even number between one and ten}*

On a Venn diagram, each set is represented by a circle which contains all the elements within it.

## Set Notation | ## Definition | ## Venn diagram Representation |

n($A$) | Number of elements in set $A$. | |

The empty set $\emptyset$ | A set which has nothing in it. | |

Universal Set $\xi$ or $E$ or $U$ | The set containing all elements. | |

Complement $A'$ or $\overline{A}$ | All elements | |

Intersection $A \cap B$ | All elements in | |

Union $A \cup B$ | All elements in | |

Conditional (A given B) $A|B$ | All elements in $A$ | |

Disjoint sets | Two sets that share none of the same elements. | |

Subset $A \subset B$ | $A$ is a set inside of $B$. |

You can use Venn diagrams to work out the probability of an event by counting the number of elements in the relevant set and taking that as a fraction out of the total.

*The Venn diagram shows the numbers of students that said they liked green and blue. *

*Find the probability that when selected at random, a pupil would like:*

*1. The colour green:*

$\text{N(Year 11 Students)}= 100$

*Work out the number of students that like green.*

$\text{n}(G)= 20+23=43$

*Work out the probability that a student likes green.*

$\text{P(}G)= \frac{43}{100}$

*2. Both blue and green:*

*Work out the number of students that like both blue and green.*

*$\text{n(}B\cap G)= 20$*

*Work out the probability that a student likes both blue and green.*

*$\text{P(}B \cap G)= \frac{20}{100}=\frac{1}{5}$*

*3. Blue given they like green:*

*Take the number of students that like both blue and green as a fraction out of the number that like green.*

*$\text{n(}G)=43 \\\text{ n(}B\cap G)= 20 \\\text{P(}B|G)=\frac{20}{43}$*

**

*No***te:** In the example above* *$25$* is the number of students that did not fit into either category - they did not like blue or green!*

A set is a collection of 'things' known as elements. In statistics, sets are usually presented on a Venn diagram. This provides an easy way for the probability of an element, inside a given set, to be worked out.

Sets are represented by a list of elements inside two curly brackets: { }

*The set of even numbers between *$1$* and *$10$*:*

*{*$2,4,6,8$*}**{*$x:$*is an even number between one and ten}*

On a Venn diagram, each set is represented by a circle which contains all the elements within it.

## Set Notation | ## Definition | ## Venn diagram Representation |

n($A$) | Number of elements in set $A$. | |

The empty set $\emptyset$ | A set which has nothing in it. | |

Universal Set $\xi$ or $E$ or $U$ | The set containing all elements. | |

Complement $A'$ or $\overline{A}$ | All elements | |

Intersection $A \cap B$ | All elements in | |

Union $A \cup B$ | All elements in | |

Conditional (A given B) $A|B$ | All elements in $A$ | |

Disjoint sets | Two sets that share none of the same elements. | |

Subset $A \subset B$ | $A$ is a set inside of $B$. |

You can use Venn diagrams to work out the probability of an event by counting the number of elements in the relevant set and taking that as a fraction out of the total.

*The Venn diagram shows the numbers of students that said they liked green and blue. *

*Find the probability that when selected at random, a pupil would like:*

*1. The colour green:*

$\text{N(Year 11 Students)}= 100$

*Work out the number of students that like green.*