the question is, does
I don't think so.
Try tossing a fair die.
I get 1/6 =1/6 for the first two
but then I get 1/6 and 1/4 on the last one.
Then the property is proved as follows (for instance):
, and the three (main) events on the right-hand side are disjoints, hence
and you can compute the latter probabilities using (and similarly for the last term) and the assumptions (recalled above).
You should get , or in fact , which is the same...