## 1.3.1 Random Experiments

Before rolling a die you do not know the result. This is an example of a **random experiment**.
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**. Thus in
the context of a random experiment, the sample space is our *universal set*. Here are some
examples of random experiments and their sample spaces:

- Random experiment: toss a coin; sample space: $S=\{heads, tails\}$ or as we usually write it, $\{H,T\}$.
- Random experiment: roll a die; sample space: $S=\{1, 2, 3, 4, 5, 6\}$.
- Random experiment: observe the number of iPhones sold by an Apple store in Boston in $2015$; sample space: $S=\{0, 1, 2, 3, \cdots \}$.
- Random experiment: observe the number of goals in a soccer match; sample space: $S=\{0, 1, 2, 3, \cdots \}$.

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. In the example of tossing a coin, each trial will
result in either heads or tails. Note that the sample space is defined based on how you define your random
experiment. For example,

Example

We toss a coin three times and observe the sequence of heads/tails. The sample space here may be defined as
$$S = \{(H,H,H), (H,H,T), (H,T,H), (T,H,H), (H,T,T),(T,H,T),(T,T,H),(T,T,T)\}.$$

Our goal is to assign probability to certain **events**. For example, suppose that we would like
to know the probability that the outcome of rolling a fair die is an even number. In this case, our event
is the set $E=\{2, 4, 6\}$. If the result of our random experiment belongs to the set $E$, we say that
the event $E$ has occurred. Thus an event is a collection of possible outcomes. In other words, an event
is a subset of the sample space to which we assign a probability. Although we have not yet discussed how
to find the probability of an event, you might be able to guess that the probability of $\{2, 4, 6 \}$ is
$50$ percent which is the same as $\frac{1}{2}$ in the probability theory convention.

Outcome: A result of a random experiment.

Sample Space: The set of all possible outcomes.

Event: A subset of the sample space.

*Union and Intersection:* If $A$ and $B$ are events, then $A \cup B$ and $A \cap B$ are also events.
By remembering the definition of union and intersection, we observe that $A \cup B$ occurs if $A$ __or__ $B$
occur. Similarly, $A \cap B$ occurs if both $A$ __and__ $B$ occur. Similarly, if $A_1, A_2,\cdots, A_n$ are
events, then the event $A_1 \cup A_2 \cup A_3 \cdots \cup A_n$ occurs if __at least__ one of
$A_1, A_2,\cdots, A_n$ occurs. The event $A_1 \cap A_2 \cap A_3 \cdots \cap A_n$ occurs if
__all of__ $A_1, A_2,\cdots, A_n$ occur. It can be helpful to remember that the key words "or" and "at least"
correspond to unions and the key words "and" and "all of" correspond to intersections.