1.4.2 Event Space

Using the sample space, we can define different events of interest. An event is a subset of the sample space, that contains the outcomes that we are interested in. If the actual outcome of the random experiment is one of the outcomes of a particular event, then we say that the event has “occurred”. It is generally denoted by any capital letter such as . Fig 1.10 shows two different events of a die. Event  is “rolling any of the first three numbers” and can be denoted by the set {1, 2, 3} while Event  is “rolling any of the even numbers” and can be denoted by the set {2, 4, 6}.

It’s worth noting that an outcome of the experiment can occur in multiple events. As shown in Fig 1.10, Outcome 2 is shared between Event  and Event .

The set of all possible events (subsets) is called an Event Space, denoted by (F-script).

Example 1.22. Let’s illustrate the relation between sample space, events and event space with a coin toss experiment (Fig 1.11):

Random experiment: Consider a simple experiment of tossing a fair coin. The coin can land either heads () or tails ().

Sample Space: This is the set of all possible outcomes of the experiment. In this case, sample space  is {, }, which represents the outcomes "heads" and "tails."

Events: Subsets of the sample space that represent specific occurrences of interest. Here are a few possible events and their corresponding subsets:

• Event : Getting either heads or tails = {, }
• Event : Getting heads = {}
• Event : Getting tails = {}
• Event : Getting neither heads nor tails i.e. empty set (impossible event) = { }

Event space: Set of all possible events. Event space is {{}, {}, {,}, { }}

Observe that the sample space is also an event (Event ) and is a member of the event space. In other words,  is a subset of , denoted as   .

Recall from the set “Membership” section (1.3.3.1), that suppose we have a set  and an element , if  is in , it is a member of , denoted by   .