what is the appropriate approach?

In a game, two players **A** and **B** shoot a target (both to the same target). Suppose that shooters are independent and the probability of (successful) shooting to target is relatively, *p**A* and **p****B**. If **A** shoots the target (successfully) before **B**, how much is it probable that **A **is the starter of the game?

PS. would you please first give your opinion about the appropriate approach and why you choose that?

probability models(distributions)

Markov chains

non-frequentist approaches like Bayesian solutions, or other things!