Probability on 2 random events combined

#1
event A and B are not related but :
if A happens then the event C has a probability W of happening.
if B happens then the event C has a probability Z of happening.

if both conditions happen A and B, Is the probability of C happening the combination of probabilities, W*Z then?
Thanks
 
#3
A and B are independent events with no relation.
For example:
If a family have children (Event A) the chances the family has a dog also (Event C) is W.
if a family has a cat ( Event B) the chances the family has a dog too (Event C) is Z.

So if we have a family with children and a cat, Event A and Event B, what are the chances the family has a dog (Event C) ?

Is it W*Z the combination ? Is it Max(W,Z) not improving having both A and B? Is it W+Z improving the chances?
Thanks