1. Probability Space
====================

Probability space is a mathematical model for random experiments with three components:

- :math:`\Omega` - The set of all possible outcomes
- :math:`\mathcal{F}` - Collection of events (subsets of :math:`\Omega`)
- :math:`\mathbb{P}` - A function that assigns a probability between 0 and 1 to each event (:math:`\mathbb{P} : \mathcal F \to [0,1]`)


**Rolling dice**

.. image:: ./images/dice.png
   :width: 250px
   :alt: Dice
   :align: center

Let's take rolling dice as an example. Probability space can be applied here to determine the probabilities of rolling,
let's say, a 6 on the next roll, or how likely are we to end up with odd number.


**Probabilities of getting 3**

.. math::
   \Omega = \{1,2,3,4,5,6\}

   \mathcal{F} = \{\emptyset, \{3\}, \{1,2,4,5,6\}, \Omega\}

   A = \{3\}, \quad A \in \mathcal F

   \mathbb{P}(A) = \frac{1}{6}

Where:
   - :math:`\Omega = \{1,2,3,4,5,6\}` - Dice has 6 sides so it contains all the possible outcomes
   - :math:`\mathcal{F} = \{\emptyset, \{3\}, \{1,2,4,5,6\}, \Omega\}` - The event we're interested in. (:math:`\emptyset`) - Indicating that no outcome satisfies the condition (:math:`\mathbb{P}(\emptyset) = 0`). :math:`\{3\}` - The outcome is 3. :math:`\{1,2,4,5,6\}` - The outcome is not 3. :math:`\Omega` - The outcome is always true (:math:`\mathbb{P}(\Omega) = \frac{|\Omega|}{|\Omega|} = \frac{6}{6} = 1`)

So, the result is :math:`\frac{1}{6}`, which is ~0.17 = ~16.7%