# Summary 1¶

So far we have seen that we can perform **experiments**, which are procedures that can be *repeated*.

We've said that these experiments can be classified into:

**deterministic experiments**: those that have a result completely determined,**random experiments**: those that have results subject to chance.

We've called **outcome** to every *result* we get when we perform an experiment, and we've organised them in a *set* called **sample space**.

The **sample space** is denoted with the Greek capital letter *Omega* $\Omega$, and every **outcome** in the sample space is denoted with the Greek lower case *omega* $\omega$.

Mathematically we write $\omega\in\Omega$.

An interesting thing we've noticed is that the *sample space* of any *deterministic experiment* has **only one element**, which is the only possible outcome of the experiment.

After that, we've defined two different types of **probabilities**:

**theoretical probability**: which is the*likelihood*of an outcome, mathematically written $P(\omega)$ for $\omega\in\Omega$,**empirical probability**: which is the*relative frequency*of an outcome, or the*proportion*of times we get that outcome.

We've explored some **empirical probabilities** and found that the more times we perform an experiment, the closer that gets to the **theoretical probability**.

At this point, we established that the probability of an **impossible outcome** is $0$, and the probability of a **certain outcome** is $1$.

With this, we realized that every probability will be **between $0$ and $1$**. Mathematically, for any $\omega\in\Omega$, $0\leq P(\omega)\leq 1$.

We then defined the concept of **event** as any *subset* of the *sample space*, and realised that we can use set operations to describe more complex events.

Also, we discovered that $P(\omega)=P(\{\omega\})$, which will allow us to think only in terms of *probabilities of events* instead of *probabilities of outcomes*.

We then introduced two very special events:

- $\emptyset$: which is the event of
*nothing happening*, - $\Omega$: which is the event of
*anything happening*

Again, we established that $P(\emptyset)=0$ and $P(\Omega)=1$.

Finally, we've learnt that the *probabilities of events that make the whole sample space* **add up to $1$**.