Given the following Question:
The possibility to win contest $A$ is $P1$, and the possibility to win contest $B$ it's $P2$.
We buy tickets for contest $A$ until we win for the first time, then We buy tickets for contest $B$ until we win for the first time, then again we but tickets for contest $A$ until we win the the first time and so on.
In other words, We try $A$ multiple times until we win once, then We go $B$ until we win etc.
Let $P(n)$ be the possibility that the $n$th ticket purchased is for contest $A$, find a recursive function for $P(n)$.
What I have done:
First, since we are talking about first win we can use the Geometric distribution.
I figured out that in order for ticket $n$ to be for contest $A$ then before that we are supposed to win even number in contest $A$ (Not sure if this helps at all)
Say probability of winning contest $A$ is $p$ and contest $B$ is $q$ (instead of using $P1, P2)$.
$P(1) = 1$
$P(n) = P(n-1)(1-p) + (1-P(n-1))q \,$ ($n \gt 1$).
I validated this for a few low values of $n$.
$P(2) = (1-p)$
$P(3) = (1-p)^2 + p q$
$P(4) = (1-p)^3 + p(1-q)q + (1-p)pq + pq(1-p) = (1-p)^3 + p(1-q)q + 2 (1-p)pq$
...