4.2.2 Probability & Venn Diagram

Venn diagram:

A Venn diagram is a graphical representation of the relationships between different sets of elements or groups.

• It consists of overlapping circles, each representing a set, with the shared area representing the elements that belong to both sets.

• The diagram can be used to illustrate the similarities and differences between sets, as well as to show how they are related to each other.

• Venn diagrams can be used in a variety of fields, such as mathematics, logic, statistics, and computer science, to visualize and analyze complex data and information.

• Drawing a Venn diagram is a simple and effective way to represent relationships between sets of data or groups.

• Here are the steps to draw a basic Venn diagram.

• Draw a rectangle that represents the entire universe of data or groups you are comparing.

• Determine the number of sets you will be comparing and draw circles or ovals for each set.

• Overlap the circles or ovals as necessary to represent the relationship between the sets.

• Label each circle or oval with the name of the set it represents.

• Add labels or shading to the overlapping areas to indicate the characteristics that the sets have in common.

Venn diagram with conditional probability:

Venn diagrams can be useful for understanding and visualizing conditional probabilities.

• Here is an example of how to use a Venn diagram to calculate conditional probabilities

Worked example:

There are \( 32 \) students in a class.

\( 5 \) do not study any languages.

\( 15 \) study German (G).

\(18 \) study Spanish S.

Complete the venn diagram to show this information.

\( 5 \) do not study any language so place \( 5 \) in the Venn diagram outside of both bubbles.

Find the intersection (GS) by finding how many students have been counted twice.

\( G+S-both+neither=32 \)

\( 15+18-both+5=32 \)

\( 38-both=32 \)

\( 38-32=both \)

\( 6=both \)

Simplify

A student is chosen at random.

Find the probability that the student studies Spanish but not German.

From the Venn diagram, \( 12 \) students study Spanish but not German.

So, the probability will be

\( P(Spanish)=\frac{12}{32} \)