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2.1.1 The sample space, ${\Omega}$

When a coin is tossed twice there are four distinct outcomes and these can be listed as {HH,HT,TH,TT}.

More generally an experiment will have ${N}$ distinct outcomes which can be denoted by ${\omega_{1},\omega_{2},\dots,\omega_{N}}$ .

The set of all possible outcomes is $\Omega=\{\omega_{1},\omega_{2},\dots,\omega_{N}\}$ and is known as the sample space. $\Omega$ is pronounced “Omega”. A particular outcome ${\omega\in\Omega}$ is a sample point. An ‘experiment’ taking place corresponds to one of the $\omega\in\Omega$ ‘happening’

Example 2.1.

a.

A die is thrown so

$\displaystyle\Omega=\{{\color[rgb]{1,1,1}\quad 1,2,3,4,5,6}\quad\}.$
b.

Three coins are thrown so

$\displaystyle\Omega=\{{\color[rgb]{1,1,1}\quad HHH,HHT,HTH,THH,TTH,THT,HTT,TTT% }\quad\qquad\qquad\}.$
c.

A die and a coin are thrown so

$\displaystyle\Omega=\{{\color[rgb]{1,1,1}\quad H1,H2,\dots,H6,T1,\dots,T6}% \qquad\qquad\}.$

The set of outcomes in the sample space has two very important properties.

The set of outcomes in the sample space are exhaustive: all possible outcomes are listed, and exclusive: no two outcomes can both occur.

The number of elements in the sample space is the number of distinct possible outcomes of the experiment. Sometimes it is helpful to represent the sample space diagrammatically, as in the following examples.

Example 2.2.

A diagram of the sample space for a die and a coin

H	*	*	*	*	*	*
T	*	*	*	*	*	*
	1	2	3	4	5	6

Each point represents a possible outcome in the sample space.

Example 2.3.

The sample space for drawing a single card from a pack of playing cards:

${\spadesuit}$	*	*	*	*	*	*	*	*	*	*	*	*	*
${\heartsuit}$	*	*	*	*	*	*	*	*	*	*	*	*	*
${\diamondsuit}$	*	*	*	*	*	*	*	*	*	*	*	*	*
${\clubsuit}$	*	*	*	*	*	*	*	*	*	*	*	*	*
	A	2	3	4	5	6	7	8	9	10	J	Q	K

Identifying the sample space correctly is extremely important because all subsequent probability calculations make use of the structure of the sample space. The examples so far have had a sample space we can write down (relatively) easily. However some sample spaces can be infinite:

Example 2.4.

A coin is thrown until a tail is observed so that

\displaystyle\Omega=\{{\color[rgb]{1,1,1}\quad T,HT,HHT,HHHT,\dots}\quad\}.

Example 2.5.

The lifetime of a computer component is measured so that

\displaystyle\Omega={\color[rgb]{1,1,1}[0,\infty)}

Even when the space is finite, listing it may present challenges:

Example 2.6.

Two cards are selected (without replacement) from a pack of ${52}$ playing cards.

There are ${52}$ ways of choosing the first and ${51}$ ways of choosing the second having chosen the first. In all ${52\times 51}$ which gives a big diagram!

Example 2.7.

A bag contains three balls, red, white and blue. A ball is selected and its colour observed, then without replacement a second is selected.

The sample space is ${\Omega=\{{\color[rgb]{1,1,1}\quad\mbox{BR, BW, WR, WB, RW, RB}}\quad\qquad\}}$ .

Example 2.8.

Two people are selected from the following four {Alf, Bert, Cynthia, Doris} and order is unimportant. List the possible outcomes.

${\Omega=\{{\color[rgb]{1,1,1}\quad\mbox{AB, AC, AD, BC, BD, CD}}\quad\qquad\}}$ .