# Distribution of color observation in Polya-like urn model

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.