Use the BINOMDIST function on your spreadsheet.

For something with a 17% chance of winning, after 5 tries:

Wins Prob
0 - 39.4%
1 - 40.3%
2 - 16.5%
3 - 3.4%
4 - 0.3%
5 - 0.01%


So probability of:

1+ wins = 60.6%
2+ wins = 20.3%