What is a Probability Space

In probability theory, a probability space is a mathematical construct that provides a formal model of a random process or “experiment”.

This space can also be defined as a probability triple $(\Omega , F, P)$, where:

• $\Omega$ is the Sample Space or the Set of all possible outcomes.
• $F$ is the Event Space or in other words a Set of outcomes in the Sample Space.
• $P$ is a Probability Function which returns for each event in $F$ a probability of it occurring, which is a number between 0 and 1.

Coin toss example

In the coin toss example we have only two possible outcomes:

• Heads (H)
• Tails (T)

Therefore the Sample Space here is represented as $(\Omega) = (H, T)$.

Considering $F=$“The coin lands on Heads” and since both events have equal probability of occurring $P = [P(H) = P(T) = 0.5]$

Card from a deck

Another typical problem is the one of the deck of cards. In a deck of cards there are 52 total cards, which as a whole represent our Sample Space $\Omega$.

Let’s say that the Event Space $F$ is the event of drawing a card of spades, let’s call the event $A$.

Therefore $P(A) = \frac{13}{52} = \frac{1}{4} = 0.25$

Sources

• https://en.wikipedia.org/wiki/Probability_space
• https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Introductory_Statistics_(Shafer_and_Zhang)/03%3A_Basic_Concepts_of_Probability/3.01%3A_Sample_Spaces_Events_and_Their_Probabilities