I am trying to calculate the expected value of an action but this action can have a probability of 20% to add another action and 80% to not add the action. with this I would start with a base number of actions being 11 but then depending on the number of extensions I get I could end with 13-14 or even 15 actions.
How would I calculate the expected value of said 11 actions and then in turn how many action would I end up with total. For a very rough and non-complete answer this would be ((11*.2) + 11) to have a base general expected total actions but would like to have a more complete answer.
I understand general expected value analysis where I calculate the distribution of values when talking about a single roll of a dice or so on what the expect value of the dice could be but dont understand how it would be calculated out when the amount of actions is effectively variable.
Example of what could happen.
I play a game where one action will always bounce 11 times at minimum but on each individual bounce it can increase the remaining amount of bounces left. With this it could then say have a total number of bounces going from 11 to 15 or even 20 if you are super lucky.
I have been searching for examples of this type of question but I have not been able to find anything calculating the expected value of this. Either how would I go about doing this or what do I need to look up to be able to learn this for myself. This may end up using limits but I can't see what exactly this type of question would require to answer.
Would this effectively just be (11 * (20%) + 11) * 20%)? Now this could be where the limit comes in because it would eventually have a limit reaching 0 to have a more defined and "Correct" answer (But it has been a long time since I have done any calculus).
if you have suggestion or a possible solution to this. I would appreciate a lamens answer first and then if you would like you can define a formal answer for me to look at.
If each action has the same expected direct value of $v$ and the same probability $p$ of generating an extra action
then the total expected value of an initial action and its descendants is $t=v+pt$ which you can rearrange as $t= \frac{v}{1-p}$
and the total expected value of $n$ initial actions and their descendants is $\frac{nv}{1-p}$
which with $n=11$ and $p=0.2$ is $\frac{11v}{1-0.2}=\frac{55}{4}v$.