I am pretty new to probabilities but have become fascinated as of late. I am trying to answer every tricky question that comes to mind and this one is stumping me.
Essentially, there is some set A which contains elements x, y and z. Event A means that the entire set is present. It has been determined that,
P ( A | x ) = 0.3 P ( A | y ) = 0.5 P ( A | z ) = 0.2
So, if we only know that x and y are present then what is the likelihood that the entire set A is present?
Intuitively it seems that the more events that are present within the set the higher the likelihood is of the entire set is present. So I was thinking it might be the union of each of the contidional probabilities for those events that are known. So, if x and y are present, the probability of set A occurring would be 0.65?
But I also concluded that since the last event z must be present for A to occur at all then the probability that A occurs might simply be the probability of A given z. Or the probability of those events that are still unknown in the set occurring since A is contingent on all of its elements occurring.
I hope I am asking this in the right way. Thank you for any help.