Probability and Relative Frequencies: Sample Spaces

Probability and Relative Frequencies: Sample Spaces

Learning Intentions

To solve problems involving sample spaces.

Success Criteria:

• Determine the size of a sample space

• Use a table to construct a sample space

• Use a systematic list to construct a sample space

Why Sample Spaces are Essential

A sample space is a list of all possible outcomes in a probability experiment. An example is the six possible outcomes of rolling one die. Each outcome represents a unique possibility, and the sample space encompasses all such possibilities.

Being able to construct a list of all possible outcomes of a probability experiment allows us to calculate the probability of an event occurring. Probability is defined as the number of favorable outcomes divided by the total number of possible outcomes. Without a well-defined sample space, accurate probability calculations are impossible.

Constructing a Sample Space

A sample space is a list of all possible outcomes. The list can be constructed using either a table or a systematic list.

Using a Table

A table is used when there are two stages in a probability experiment. The outcomes of one stage are listed across the top, and the other stage is listed along the side. The table is then filled with the combinations of the two trials. This method is particularly useful when visualizing all possible pairings of outcomes.

Example: Tossing a coin and rolling a die.

Size of Sample Space

The size of the sample space is determined by the product of the number of outcomes in each stage. This is based on the fundamental counting principle, which states that if there are m ways to do one thing and n ways to do another, then there are m \times n ways to do both.

Example:

A die has 6 outcomes {1, 2, 3, 4, 5, 6} and a coin has two outcomes {H, T} . Therefore, the size of the sample space = 6 \times 2 = 12

Using a Systematic List

A systematic list is constructed by systematically listing all the outcomes of the first stage with each of the outcomes of the second stage in turn. This ensures that every possible combination is accounted for, reducing the risk of missing an outcome.

Example:

A coin is tossed and a spinner is spun with the outcomes A, B, C on each selection. The sample space would be:

{(H, A), (H, B), (H, C), (T, A), (T, B), (T, C)}

Selecting Without Repetition

If selecting two items from a list, repeats cannot happen. This means once an item is selected, it cannot be selected again for the same combination.

Example:

Colby, Sharlah, Bridget, and Tom nominate for the school tennis team. Only two positions are available. The possible team combinations would be:

{(Colby, Sharlah), (Colby, Bridget), (Colby, Tom), (Sharlah, Bridget), (Sharlah, Tom), (Bridget, Tom)}

Examples

Example 1: Using a Table

Nicole is packing for an overseas holiday. She plans on taking:

• 3 shirts that are blue, white and black

• 2 pairs of trousers that are tan and black

a) If each combination of shirt/trousers is an outcome, calculate the size of the sample space.

b) Use a table to construct a sample space of the different outfit combinations.

Solution to a): \text{size of sample space} = 3 \times 2 = 6

Solution to b):

Example 2: Using a Systematic List

A pizza restaurant offers two types of base (thin or thick) and two types of sauce (BBQ or tomato). These can be ordered in any combination.

a) Calculate the size of the sample space (the number of pizza combinations).

b) Make a systematic list of the pizza combinations.

Solution to a): \text{size of sample space} = 2 \times 2 = 4

Solution to b):

{(Thin, BBQ), (Thin, Tomato), (Thick, BBQ), (Thick, Tomato)}

Example 3: Using a Table without Repetition

Patrick is in the gym and wants to add weights to the bench press to increase the load. He can choose from 2kg, 5kg, 10kg or 20kg weights. There is only enough space to add two weights to the machine, and only one of each of the weights to choose from.

a) Calculate the size of the sample space.

b) Use a table to construct a sample space of the different weight combinations.

Solution to a): The possible combinations are (2, 5), (2, 10), (2, 20), (5, 10), (5, 20), (10, 20). Thus, the size of the sample space is 6.

Solution to b):

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