Stuffed Animals

This is a user suggested problem. The TopsOJ staff thanks and gives full credit to vsarca for their contribution.

I am making stuffed animals from bodies, heads, and tails. I can only operate on one animal body at a time, and I may not operate on another animal body until I finish my current animal. I create animals by flipping a coin; when the coin flips heads, I add a head if my current animal does not already have a head, and when it flips tails, I add a tail if my current animal does not already have a tail. An animal is completed once it has a head and a tail, at which point I set it aside and begin another one.

If my expected number of completed animals after 15 coin flips is \frac{p}{q}, where \gcd(p,q)=1, find p+q.

Note: it is STRONGLY RECOMMENDED by the problem author to find the closed form first, as this was the original problem statement.