2.1.2 Venn Diagrams

Venn Diagrams

A Venn diagram is a way to illustrate events and are particularly useful when outcomes overlap
Venn diagrams are mostly used for 2 or 3 events
A Venn diagram consists of a box (rectangle) and a bubble (circle/ellipse) for each event
- Bubbles may or may not overlap
- Bubble(s) is not a technical term, but we like it!
The box represents all outcomes
- It is often referred to as the Universal Set and is commonly labelled with the symbols $E$ (Kunstler script font uppercase letter E) $S, U or ξ$ (Greek lowercase letter Xi)
- There is no standardised symbol for this purpose
Bubbles are labelled with their event name (A, B, etc)
The numbers inside a Venn diagram (there should be one in each region) will represent either a frequency or a probability
- In the case of probabilities being shown, all values should total 1

2-1-2-cie-fig1-basic-venns

2-1-2-cie-fig2-various-venns

40 people were surveyed regarding which games consoles they owned.

8 people said they owned a Playstation 5 ( $P$ ) and an Xbox Series X ( $X$ ).

5 people said they owned neither of these consoles.

Of those people that owned only one games console, twice as many owned an Xbox Series X as a Playstation 5.

One of the 40 people is chosen at random. Find the probability that this person

(i)

owns both consoles,

(ii)

owns exactly one console,

(iii)

doesn’t own a Playstation 5.

2-1-2-cie-fig3-we-solution

The rectangle in a Venn diagram is a key part of the diagram
- it represents all possible outcomes of the experiment
- the bubbles merely represent the events we are particularly interested in
- there is usually a few possibilities that fall outside of these events so this would be the section outside the bubbles but inside the rectangle
A quick ‘mini-Venn’ diagram shading the parts required to answer the question can be useful rather than always drawing a full Venn diagram with all its values