7.01 Sample space

Sample space

The sample space, sometimes called an event space, is a listing of all the possible outcomes that could arise from an experiment.

For example

• tossing a coin would have a sample space of $\left\{\text{Head},\text{Tail}\right\}${Head,Tail}, or $\left\{H,T\right\}${H,T}
• rolling a dice would have a sample space of $\left\{1,2,3,4,5,6\right\}${1,2,3,4,5,6}
• watching the weather could have a sample space of $\left\{\text{sunny},\text{cloudy},\text{rainy}\right\}${sunny,cloudy,rainy} or $\left\{\text{hot},\text{cold}\right\}${hot,cold}
• asking questions in a survey of favourite seasons could have a sample space of $\left\{\text{Summer},\text{Autumn},\text{Winter},\text{Spring}\right\}${Summer,Autumn,Winter,Spring}

Notice how the the sample space is listed inside curly brackets, with a comma between each outcome. 

 

 

Practice questions

QUESTION 1

QUESTION 2

 

2D grids

One way to show the same place of a multi-stage event is using a 2D-Grid. To construct a 2D grid, we draw a plane with two axes, where each axes represents one of the events that are taking place. Then we put a dot in each position to represent a possible outcome. 

For example, in the grid below, the vertical axis represents the possible outcomes of tossing a coin, and the horizontal axis represents the possible outcomes when rolling a die:

Each dot represents an outcome when rolling a die and tossing a coin. So the dot in the top left corner represents the outcome of rolling a $1$1 and tossing a Heads.

Worked example

Two six-sided dice are rolled at the same time. 

(a) Display the sample space as a 2D grid.

(b) Find the probability of rolling a double.

(a) Two events are happening in this example, where each event is rolling a die. Therefore our 2D grid needs to have both axes representing rolling a die without outcomes $1-6$1−6 on each axes:

(b) The dots that represent rolling a double are highlighted below:

$P(double)$P(double)	$=$=	$\frac{6}{36}$636​
	$=$=	$\frac{1}{6}$16​

Practice question

QUESTION 3