Sample Space

Before rolling a die you do not know the result. This is an example of a random experiment or trial. In particular, a random experiment is a process by which we observe something uncertain. After the experiment, the result of the random experiment is known.
An outcome is a result of a random experiment. The set of all possible outcomes is called the sample space, usially denoted by Ω.
In the context of a random experiment, the sample space is our universal set. When we repeat a random experiment several times, we call each one of them a trial. Thus, a trial is a particular performance of a random experiment.
Example: Here are some examples of random experiments and their sample spaces:
Random ExperimentExperimental Outcomes
Toss a coinHead, Tail
Conduct a sales callPurchase, no purchase
New born babyGirl, Boy
Roll a die1, 2, 3, 4, 5, 6
Hold a oparticular share of stock for one yearPrice of stock goes up, price goes down, no change in stock price
Observe the number of goals in a hockey match1, 2, 3, 4, ...
Table cell 3Table cell 4
The following codes can be used to simulate tossing a fair coin.
Example: Consider the experiment of tossing a fair coin three times and counting the number of heads (H) and tails (T). There are eight possible outcomes:
\[ \Omega = \left\{ HHH, \ HHT, \ HTH, \ THH, \ THT, \ TTH, \ HTT,\ TTT \right\} . \]
So its cardinal number is |Ω| = 8.
Example: Consider the experiment of tossing rolling two balanced dice when they are clearly idenitified as different. The outcome set consists of 36 possible pairs of numbers from [1..6]:
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
Let the event ω consists of all rolls with a total of 5 points. Then
\[ \omega = \left\{ (1,4), \ (2,3), \ (3,2), \ (4,1) \right\} \]
and its cardinal number is |ω| = 4.    ■