Event

From Sample Space to Event

The sample space $S$ of a random experiment is the set of all possible outcomes. But when we actually perform the experiment, we are rarely interested in every outcome individually. Instead, we care about whether a particular happening occurs — for example, "getting exactly one head" when tossing a coin twice.

Consider the experiment of tossing a fair coin two times. The sample space is

Now, which outcomes correspond to "exactly one head"? Only $HT$ and $TH$. These two outcomes together form a set:

Notice that $E$ is a subset of $S$. This is not a coincidence — it is the central idea of the chapter. Every event we can describe in words corresponds to some subset of the sample space. The table below shows this correspondence for the coin-tossing experiment.

The last two rows are particularly instructive. The event "number of tails is at most two" is always true — every outcome satisfies it — so its corresponding subset is the whole sample space $S$. The event "number of tails is more than two" is impossible — no outcome satisfies it — so its corresponding subset is the empty set $\phi$.

Formal Definition of an Event

Based on the discussion above, we define an event as follows.

This definition is deceptively simple. It means that the language of set theory — subsets, unions, intersections, complements — becomes the language of probability. When we say "event $E$ occurs", we mean that the outcome of the experiment is an element of $E$. When we say "event $E$ does not occur", we mean the outcome lies in the complement $E' = S \setminus E$.