Two-Way Tables, Sample Space Diagrams, Tree Diagrams and Venn Diagrams

Two-Way Tables, Sample Space Diagrams, Tree Diagrams and Venn Diagrams

Two-Way Tables

  • Two-way tables, also called contingency tables, are used to represent bivariate frequency data. These tables present outcomes from two variables, each having two or more categories.

  • These tables are helpful in identifying patterns, trends, and relationships between two variables.

  • The totals in each row and column are important for calculating probabilities.

  • When using a two-way table to calculate probability, an event’s probability is the frequency of that event divided by the total frequency.

Sample Space Diagrams

  • Sample space diagrams are graphical representations which allow visualisation of all possible outcomes of an experiment or event.

  • A sample space diagram is particularly useful in representing outcomes resulting from two or more combined events.

  • Each outcome is represented separately in the sample space diagram, and the complete set of distinct outcomes is known as the sample space.

  • The probability of any event can be calculated by counting the number of desired outcomes and dividing by the total number of outcomes.

Tree Diagrams

  • Tree diagrams are a graphical way of listing all possible outcomes from two or more events.

  • They’re particularly useful for visualising multiple stage experiments where each stage depends on the previous one, otherwise known as dependent events.

  • The branches of a tree diagram represent different possible outcomes, and their length is proportional to the probability of the outcome.

  • The sum of the probabilities from each set of branches should equal 1, based on the law of total probability.

Venn Diagrams

  • Venn diagrams are visual tools used in probability to show the relationship between sets of events. Circles in a Venn diagram represent events, with the area inside the circle representing the occurrence of the event.

  • Intersections between circles represent the occurrence of multiple events simultaneously, also known as the intersection of events.

  • The union of events is represented by all the areas mutually covered by the circles.

  • The area outside all circles but inside the rectangle (which represents the sample space) represents the complement of an event, showing the outcomes that do not belong to the event.

  • Venn diagrams are particularly useful for visualising probabilities related to combined events and for calculating conditional probabilities.