A few layers to get through to get to what would be a plausible scenario in draw odds, but each choice on an application acts as its own separate application because Johnny's three choices will be looked at before Bill's.
First hurdle: your app number has to be somewhere in the mix for total tags for that species you put in for.
Second hurdle: your app has to be high enough in the pecking order (looked at sooner than others) to even have your choices matter.
Third hurdle: your choices on your app have to be high enough on the list against the other choices on other apps for it to matter.
Best thing anyone can do is to look at how quickly a hunt "fills up" based on how many tags are awarded to 2nd and 3rd choice, which are usualy the "least desirable". The more desirable hunts you put in for your choices, the lower your chances are at getting drawn.
Oryx app with 20,000 tags available out of total of 200,000 applications. You have a 1/10 chance your app will be in the mix.
Same app with 50 tags for each choice and 1,000 put in for each of your choices. You have a 1/20 chance for your choices to win a tag.
A Monte Carlo method can't be used to predict the number of times a hunt code will show up with good odds because the applications are only shuffled once.