There are two variables $a$ and $b$.
(The first value of variables : $a=1,b=0$)
We will add $1$ to $a$ or $b$ for $n$ times. ($n \in \mathbb N$)
The probability of adding to each variables are :
$$P_a = k({a \over a+b})+(1-k)({b \over a+b})$$
$$P_b = k({b \over a+b})+(1-k)({a \over a+b})$$
($P_a$ : probability of add 1 to a, $P_b$ : probability of add 1 to b)
$k$ is constant number in $(0,1)$.
I want to find $P(a=1)$,$P(a=2)$,...,$P(a=n)$.
I think this discrete probability distribution is similar with binomial distribution. However, each sequences are not independent so I am really confusing what this probability distribution is.