Hmm labeled according to the turn in which you would win, where the part in brackets can be permuted

2) [WW]W

3) [WWLL]W

4) [WWLLLL]W

...

Probability for (2) should be 1/C(20,3).

Probability for (3) should be C(3,2)*C(17,2)*4!/P(20,4)*1/16. (According to standard order of operations; parentheses are not needed.)

where C(n,k) is number of k-subsets of {1,2,...,n} and P(n,k) is number of k-permutations of {1,2,...,n}.

There may be an easier way.. but following the same pattern it's easy to write terms (4)-(10).