Thanks for contest and prizes, was very funny!

Username: Akai
Code: XMAS

The probability is 2/3....in fact the thing that you pick a BTC coin don't change probability of the choose of the box.
If for example we have:
Box 1: BTC 1 -btc2
Box 2: ltc 1 -ltc 2
Box 3: BTC 3 - ltc 3

The extracted BTC could be only one of BTC 1, btc 2, BTC 3, so possible cases for the BTC are 3/3.
If I pick one of 3, the P is now 2/3.