Pretty sure the number of applicants is every hunter regardless of whether they are in a party. If there's 10 tags available they pull the first name/party (assigned by random number), if it's a party they issue that many, repeat until the available tags are all issued. If the last name drawn is a party and there's more people in the party than available tags remaining they issue tags to everyone in the party and exceed the quota by 1 or a few tags.
The odds would practically be 50%, but the actual odds would be difficult to calculate because that would depend on how many parties make up the 20 applicants complicated by the random number of tags issued for each name/party pulled out of the hat.
Example using Google AI:
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