Game with 5+5 doors

[jason]

Two friends are sitting at the two opposite sites of a building. Each of them has 5 closed doors in front of him. They toss a fair coin and each time they get a head, they open one of their own doors, otherwise they do nothing. What is the probability that all 5 doors of each friends are opened after the same number of tosses?

Negative binomial distribution? But how can we apply it when we have two discrete players, whose coin tosses are independent?

Any other ideas?

Btw this is not school homework or anything - just for fun!

[Guest]

Is the answer 1/2 ?

[jason]

I don't know the answer but for sure it is not 1/2. It must be a very small number (by intuition).

We are looking for something like this:
If H = heads and T = tails, the sequence of outcomes must be like this: (just an example)
Player A: T H T T H T T T H H T H
Player B: H T T H H T T T T H T H

i.e they both finish after the same number of tosses.
Obviously the last toss is H for both (otherwise they would keep on tossing).
I don't know how to continue.