1.4.1 Sample Space

To keep track of all the possible outcomes for a given random experiment, we use the concept of a sample space. The sample space is essentially the set of all possible outcomes that could occur for that experiment. It is generally denoted by  (Omega) or letter . Fig 1.9 shows the sample space for an experiment of rolling a die. Considering the six outcomes of the die, the sample space can be denoted as  = {1, 2, 3, 4, 5, 6}. To avoid any clutter, only two outcomes have been highlighted in Fig 9.

Sample spaces can take on various forms depending on the nature of the random experiment being considered. Here are some common types of sample spaces:

Finite sample spaces: In this type, the set of possible outcomes is finite and countable, and each outcome is distinct. This means that you can list all possible outcomes explicitly. For example, when rolling a six-sided die, the sample space consists of the numbers, denoted as  = {1, 2, 3, 4, 5, 6}. Each of these numbers represents a distinct outcome of the experiment. Fig 1.9 shown earlier is a representation of this type of sample space. Another example could be flipping a coin once, where  = {Heads, Tails}. Each of these outcomes is countable, and there is a finite number of them.

Countably infinite sample spaces: In this case, the set of possible outcomes is infinite and countable, yet each outcome remains distinct. This situation involves repeating a random experiment indefinitely until a specific condition is satisfied. For instance, consider an archer aiming to hit the bullseye—our sample space comprises the number of retries required before successfully hitting the bullseye. So  = {0, 1, 2, 3, 4 . . .}. The archer can hit the bullseye in 0 retries or 100 or 1000 or maybe a million if it’s me, we can’t be sure how many retries are needed so we say that  is countable but can go on to infinity.

Uncountably infinite sample spaces: In this type, the sample space involves outcomes that are so numerous and continuous that they cannot be systematically listed or counted individually. The concept is essential in scenarios where outcomes form a continuum and can't be captured by discrete enumeration. Imagine you're measuring the duration required for a webpage to load, and the highest acceptable page load time is 3 seconds. In this scenario, the possible outcomes could be anywhere from 0 to 3 (excluding 0), depending on the level of precision. So, in this case,  = {all real numbers  such that 0 <  < 3}.  is uncountably infinite as you can measure the outcomes up to nth precision, you can say the webpage load time is 1.25 seconds or even 1.2588888888998987673 seconds if you've got a microscope for a stopwatch!

It is important to note that not all sets qualify as a sample space. The set of a sample space must exhibit qualities of being exhaustive and exclusive. The term exhaustive implies that the sample space set contains all possible outcomes. If even a single outcome is not part of the set, the set is not considered a sample space. The term exclusive suggests that the sample space contains unique elements, avoiding the repetition of elements.

Say, for the experiment of rolling a die:

• The set {1, 2, 3, 4} is not a sample space as it's not exhaustive
• The set {1, 1, 2, 3, 4, 5, 6} is not a sample space as its not exclusive

Note that the concepts of a universal set and a sample space are related but used in different contexts within mathematics. A universal set represents the entire collection of elements under consideration, while a sample space specifically refers to the set of all possible outcomes in a probabilistic experiment or event. The universal set is more general and can encompass various topics, while the sample space is specific to probability theory and random experiments.