When using (nominal) categorical data, is a classifier needed? Or is knowing the distribution enough? I am working with 2 categorical variables associated with the condition of an animal as it goes into an animal control facility, and the outcome of the intervention. There are 6 types of intake conditions and 9 types of outcomes, so I started modeling with 2 discrete random variables
$X$: Intake Type {Abandoned, Euthanasia Request,...}, and $Y$: Outcome Type {Adoption, Died, Disposal,...}.
I have about 37,000 rows of data (one row for each intake/intervention) so I calculated the empirical probability mass function (pmf):

Now let's suppose a new animal comes in. Given its condition, does it make sense to build a decision tree to classify the outcome?
Or would a calculation such as "given $X=x$, find the value $y$ which maximizes $P(\{Y=y\} \cap \{X=x\})$" provide an equivalently good answer? It just seems that if I have the pmf, it's easy to say that, for example, if the animal was an 'owner surrender' ($X=\mbox{owner surrender})$ then the best guess of the outcome is $Y=\mbox{transfer}$ since $P(\{Y=\mbox{transfer}\} \cap \{X=\mbox{owner surrender}\})$ is the largest probability ($0.189905$) in that column. It seems that a decision tree is not really needed for this problem.

Thank you Christian Hennig and K. Hencken for your comments. I will try to combine your insights into this answer, but if you wish to answer directly please feel free to do so and I will accept your answer.
Finding the value of $y$ which maximizes $P(\{Y=y\}\cap\{X=x\})$ given $x$ is a useful classifier for this problem. In fact, the proposed classifier is called a Maximum A Posteriori (MAP) classifier. A decision tree for this problem would probably not make too much sense. Answering the question 'is there a better classifier for this problem?' would first require defining a different loss function. For example, if the $y$s are assigned different weights, the optimal decision may be different from the current answer.
