Use of Diagrams and Tables to Calculate Probabilities

We can use different visual and tabular methods to help list outcomes, define events, and calculate probabilities. These methods include:

 Venn Diagrams

Venn diagrams are used to represent sets of outcomes and their relationships visually.

In the diagram above, we have two events $A$ and $B$ within the sample space (or universal set) $S$. Sometimes, the sample space is denoted by $\sigma$ or $\xi$ instead of $S$. Colored regions in this Venn diagram represent the following events:

• Green and purple regions : $A$,
• Blue and purple regions : $B$,
• Purple region : $A \cap B$,
• Green, purple, and blue regions : $A \cup B$,
• Yellow region : $\overline{A \cup B}$, alternatively, $(A \cup B)’$.

 Tree Diagrams

Tree diagrams show all possible outcomes of multi-stage events in a branching structure.  

• Each branch represents an outcome at a stage.
• 
• Probabilities are written on branches.
• 
• To find the probability of a complete path (outcome sequence), multiply along the branches.
• To find the probability of an event, add up probabilities of relevant paths.
• \(\text{Path probability} = \text{product of branch probabilities}\)
• 

 Sample Space Diagrams

A sample space diagram lists all possible outcomes in an organized way. It is especially useful for:

• Rolling two dice (as a grid)
• Tossing multiple coins

It helps identify outcomes satisfying certain conditions and makes counting outcomes easier.

Example grid for two dice:

 Tables of Outcomes

Tables can be used to display all combinations of outcomes, especially for numerical results (e.g. sum of dice).

• Rows and columns represent possible values of different variables.
• Each cell shows an outcome or result (e.g. sum of two dice).

You can use the table to:

• Count favourable outcomes
• Calculate probabilities: \( P(A) = \frac{\text{number of favourable outcomes}}{\text{total outcomes}} \)