# Understanding Random Experiment

I am self-learning probability and statistics and I am trying to understand probability at a deeper level. As part of my learning so far, I understand that

• An event is a set of outcomes of a random experiment.

Now, I am working through a problem which roughly states:

Let A be the event that a person has a disease and B be the event that a test for the disease is positive, indicating the person tested has the disease. Then there are bunch of conditional probabilities given involving A and B and I need to solve for something.

My Question is:

Since A and B are events, and there are some conditional probabilities involving for both of them, what is the random experiment for which A and B are the events?

It's a random choice among the possible outcomes of the cartesian product of the domains of events $$A,B$$, which is ($$D$$: has disease, $$T$$: test positive): $$\mathcal{S}=\{(D,T),(D,T'),(D',T),(D',T')\}$$ Here, event $$A$$ is the set $$\{(D,T),(D,T')\}$$ and event $$B$$ is the set $$\{(D,T),(D',T)\}$$.
Practically, you can think of an experiment where you first choose a random number, $$u$$, in $$U[0,1]$$ and let the person has the disease if $$u and not o/w. Following this, you'll take another sample from $$U[0,1]$$ and compare it with $$P(T|D=d)$$ similarly, where you have your sample from the set above, $$\mathcal{S}$$.