I have four scenarios with five rounds each where an event x, y, or z will be chosen.
In the first scenario, each round has a equal chance of triggering event x and will continue triggering event y each round until event x is triggered. Example1: if event x is triggered on r1, y=0. Example2: if event x is triggered on r3, y=2. Example3: if event x is triggered on r5, y=4.
Your description of the problem is unclear. If y is an event, what kind of event is it? Is it something that either happens or doesn't? If that is the case, why do you give an example with y = 4? Are you using "y" to denote both the event and the total number of times that it happened? (If so, stop doing that!) Are x,y and z mutually exclusive events? Or is it the case that several of these events can happen on the same round?
I am trying to learn how to calculate the average on the amount of times event y will trigger in each scenario
It isn't clear what you count as a "time". Does event y happen or not happen on each round? If it happens on rounds 2,4 and 5, do you count that as three times that it happened? -or do you count that as one time since it did happen at least once in the scenario?