13B - Representing sample spaces

Multi-stage events

Often we are interested in the probabilities arising from a multi-stage experiment; that is, and experiment that can be considered to take place in two or more distinct stages. Common examples of multi-stage random experiments include:

• Flipping two coins
• Rolling two dice
• Flipping a coin and rolling a die

Visual representation

Visual displays can be used to represent the sample space for more complex multi-stage random experiments in probability. We will examine the following visual representations:

• Venn diagrams
• Tables and grids
• Tree diagrams
• Karnaugh maps

The notes below provide a brief summary of each representation, please click on the links above, or on the headings below, for more information on each display.

Venn diagrams

Venn diagrams can be used with two or more events (subgroups), each of which is represented by a circle. The entire sample space is, generally, represented by a rectangle. The following Venn diagram shows Venn diagrams for 2 events and 3 events:

For more information on Venn diagram click here.

Tables and grids

Tables and grids (also known as arrays) can be used to show the outcomes in a sample space with only two events. If you need to represent more than two events you need to use a tree diagram. The following table show the sample space for flipping and coin and rolling a die:

For more information on tables and grids click here.

Tree diagrams

Tree diagrams can be used to show the outcomes in a sample space containing two or more events. If you only need to represent two event you may also consider a table or grid. When using a tree diagram remember:

• Multiply probabilities along the branches.
• Add probabilities down the tree.

Helpful hints:

• Keep probabilities as unsimplified fractions so that you can check they all add up to 1. When you answer the question you should then simplify your answer!
• If the probabilities of the branches in the same "stage" are identical up and down the tree then you are dealing with independent events. If the the probabilities are different within the same stage the events are dependent.

For more information on tree diagrams click here.

Karnaugh maps

Probabilities associated with two compound events are sometimes able to be calculated more easily using a probability table (Karnaugh map). A general map is shown below:

When using a Karnaugh map remember:

• The columns and rows add up to give the total probability for the event described.
• The final column and final row will add up to 1 (the probability of the sample space).
• You can use the additional rule for probability to help solve problems.

For more information on Karnaugh maps click here.

13B - Exercises:

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