Sample Space Diagrams

Definition of Sample Space

The sample space is the set of all possible outcomes of a probability experiment. It represents every result that could happen when an event is performed.

For example, when tossing a coin once, the sample space is {Heads, Tails}. This means the only possible outcomes are either Heads or Tails.

The sample space is important because probabilities are calculated based on the number of outcomes in this set. Every event is a subset of the sample space.

Sample Space Diagrams

A sample space diagram is a visual way to list all possible outcomes of an experiment clearly. It helps to organise and display outcomes so that none are missed.

Sample space diagrams often use tables or grids, especially when there are two or more stages or objects involved. Each cell in the table represents one possible outcome.

For example, when rolling a 6-sided die and tossing a coin, a sample space diagram can be drawn as a table with the die results as rows and the coin results as columns.

Using sample space diagrams ensures that all outcomes are listed systematically, making it easier to calculate probabilities.

Simple Probability Using Sample Spaces

The probability of an event is found by dividing the number of favourable outcomes by the total number of outcomes in the sample space.

This is written as:

$$\text{Probability of event} = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$$

Probability values always lie between 0 and 1, where:

• 0 means the event cannot happen
• 1 means the event is certain to happen

For example, if you roll a fair 6-sided die, the probability of rolling a 4 is $\frac{1}{6}$ because there is 1 favourable outcome (rolling a 4) and 6 possible outcomes in total.

Using sample space diagrams helps to count the total and favourable outcomes accurately.

For instance, consider tossing two coins. The sample space diagram lists all possible outcomes:

Coin 1	Coin 2	Outcome
Heads	Heads	HH
Heads	Tails	HT
Tails	Heads	TH
Tails	Tails	TT

There are 4 possible outcomes in total.

If the event is "getting exactly one Head," the favourable outcomes are HT and TH, so 2 outcomes.

Therefore, the probability is:

$$\frac{2}{4} = \frac{1}{2}$$

Example (inline): What is the probability of rolling a 3 on a fair 6-sided die? There is 1 favourable outcome (rolling a 3) and 6 possible outcomes, so the probability is $\frac{1}{6}$.