Bernouilli trial with variable number of experiments

Question

I'm kinda stuck on a probability problem I encountered in designing a game. Here is its description :

I'm calculating the number of turns (Tf) before a integer variable (A) reaches 0. Each turn, A decrease by 1, and has a p probability to decrease by 1 more.

Do you think that we could calculate Tf using an expression?

Thanks

Answer 1

I guess you can say that $\lceil\frac{A}{2}\rceil \le TF \le A$.

Also on each turn, the expected decrease of $A$ is $1+p$. So $E[Tf]=\frac{A}{1+p}$.

There is a bit of ambiguity in your question, as to what happens when $A=1$. Is the game over, or does play continue with the possibility of $A$ going below $0$?