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