what is probability space https://en.wikipedia.org/wiki/Law_of_total_probability
I was going through the definition  of total probability theorem and came upon this term.Can you guys tell me what it is and how to identify it and it's relation with the theorem.
 A: You define a probability space $(\Omega,\mathcal{F},\mathbb{P})$ by answering three questions:


*

*What are the possible outcomes of our "experiment"?

*How do we group these outcomes into "events" that we can assign probabilities to?

*How do we calculate the probability of each "event" (i.e., set of outcomes)?


The first question defines the sample space ($\Omega$). The second defines the set of events $\mathcal{F}$, also called a sigma-field of events. The third will define the probability measure ($\mathbb{P}$) that assigns probabilities to each set in $\mathcal{F}$.
$\mathbb{P}$ (in the form of distributions) is what most intro probability courses focus on -- Bernoulli, Binomial, Poisson, Normal, Gamma etc. $\Omega$ is usually pretty obvious (e.g., the outcome of tossing a coin $n$ times) or ignored (e.g., when you talk about a "normally distributed random variable with mean 0, variance 1). $\mathcal{F}$ is a technically difficult area that, for most applications of probabiltiy, is not an issue (e.g., we usually care about intervals on the real line or sets of integers...all nicely behaved for the most part).
The only relation with the theorem is that it provides $P()$ in the theorem and defines what types of events you can partition over to get the "total probability".
A: Loosely speaking, a probability space is a set where all your desired events live. This language is usually used in the measure-theoretic style of probability theory.
