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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?

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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 $u<P(D)$ 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, $\mathcal{S}$.

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    $\begingroup$ Thank you! There were a few gaps in my understanding. I was thinking of a RE as only containing a single step, like tossing a coin, but now i understand that it could by multi-step too. For example, drawing a card after throwing a dice. $\endgroup$
    – gmatharu
    Commented Jul 12, 2020 at 8:21

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