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.