Sample space

An experiment is an activity or procedure that produces distinct possibilities, called outcomes.

In probability and statistics, randomness is involved: outcome of experiment is uncertain.

The set of all outcomes is called the sample space. It is denoted as $\Omega=\{...\}$. Therefore, outcome a is an element of the sample space: $a\space \in\space \Omega$

Let’s see some examples:

• Experiment 1: We throw a coin. What is the sample space? Assuming that we will never see the coin land on its rim, there are two possible outcomes: heads and tails. We therefore take as the sample space associated with this experiment the set $\Omega = \{\text{Heads, Tails}\}$
• Experiment 2: We throw a die. What is the sample space? We suppose it is a normal die, that is, it has 6 faces. As a result, the sample space is $\Omega = \{\text{1, 2, 3, 4, 5, 6}\}$
• Experiment 3: Waiting time for the next bus. What is the sample space? Maybe we arrive and the bus is already there, so we don’t have to wait. Or maybe the bus never arrives at our bus stop. As a result, the sample space is $\Omega = [0, \infin)$
• Experiment 4: Output of 4 calls o random number generator (0 or 1). What is the sample space? $\Omega = \{\text{(0,0,0,0), (0,0,0,1),...,(0,1,1,1),(1,1,1,1)}\}\\ \hspace{1.3cm}=\{(x_1, x_2,x_3,x_4)\space : \space \in \space \{0,1\}\}$

<aside> 📝 We throw one die twice. The set of possible outcomes of this experiment is given by…

• Answer </aside>

Set Theory

Scientists from all over the world use the same language to talk about collections of objects. This language is known as Set Theory.

A set is simply a collection of objects. For example:

• A set of cars

  

• A set of blue cars

  

The members of a set are called elements. So, a blue car is an element of the set of all cars. Written down in the notation of the set theory, it would look like this:

Another example is that you are an element of the set of all people in the world.

More abstractly written down, look at the following set $A$, with distinct elements $a_1$ till $a_5$: