Set of events(F) Vs Set of Possible Outcomes

Hi Team

I have doubt regarding Set of events Vs Set of possible outcomes.In Week 18,“Probability Recap”,sir mentioned below points

Omega : Set of Possible outcomes

F: Set of events

p:maps events to probability

If Die has only 3 faces(1,2,3) and is rolled twice,I am interested in events where no 3 has occurred in outcome with order ignored

omega :(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)

F:(1,1),(1,2),(2,2)

p :How should I calculate my probability of each event.Below is my calculation

(1,1) —1/9

(1,2) --2/9

(2,2) --1/9

Probabilities has to be calculated from set of all possibile outcomes(Omega) and not F

my random variable X :Sum of outcomes and below is the PMF

      2 -→1/9

      3 -→2/9

      4 -→1/9

But above is not summing up to 1.Instead if I say that my event space(F) has only 4 outcomes and modify the probabilities as below ,ie.,calculation prob from F and not Omega,r.v X sums to 1

(1,1) ---¼

(1,2) --2/4

(2,2) --¼

But above cant be true as prob of getting (1,1) for 3 faced die is 1/9.

And I am not sure if sir has discussed probability calculations for an example where Omega is not equal to F.

Please let me know If am missing anything

Regards

Subbu

Hi,
I feel you’re right at this point:

Yes, you get it right:

But look from an other perspective, that 3 is anyway included in each of these sets, if we look at F
This way, we can only have the values on dice: (1,1) (1,2) (2,1) (2,2)
Thus total no. of outcomes would be 4 right?
Let me know if i made any mistake.

Thank you for reply.So are you making the point that

“Probabilities need to be calculated from the selected event space F instead of Omega”

In that case prob of each event will not be 1/|Omega|,instead it should be 1/|F|. But we frequently see in videos ,prob of each event as 1/|Omega|.

Also If we are interested to have F as only one event from Omega,we cannot say probability of occurring the event in F is 1.

Point I am trying to make here is random variable has to be created on exhaustive sample space Omega but not on any subset(F) of Omega. Please help me to understand here. Apologies if I am missing something very basic here

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