Consider an urn containing balls of $K$ different colors with $a_i$ balls of colour $i$. According to Wikipedia, if we sample $n$ balls in a Polya's fashion, the distribution of the observed colors will follow a Dirichlet multinomial distribution with parameters $n$, $(a_1, ..., a_K)$.
I am personally interested in the distribution of the observed colors in a slightly different setting. Consider the same initial urn, but now, when a ball of color $i$ is observed, with probability $p_j$ it is returned together with a ball of color $j$. Otherwise, it is returned together with a ball of the same color. Notice $p_j$ does not depend on the color of the observed ball.
I would like to know if someone studied this problem and if there are some accesible references or books I could look at for guidance. In particular, I hope the distribution of observed colors in the second setting to still follow a Dirichlet Multinomial with different parameters, even though I do not believe this is the case.