I’d say you have at least a 68.75% (11/16) chance of winning against yourself. Because you can make “stupid” moves when helping yourself win, right?
So black can win against white in three moves if they go like this:
1. W: Pawn to F4 (or F3)
2. B: Pawn to E5 (or E6)
3. B: Queen to H4
White needs to win the first coin toss (probability 1/2) and black needs at least two of the remaining four (probability (1/2)^5 * 11 = 11/32 ).
The same goes for black winning against white, so the first toss decides which side will try to win.
The probability then would be 2 * 11/32 = 11/16 = 68.75%. At least that much, because I haven’t explored all possibilities.