1. 心存侥幸地；怀着渺茫希望地
If you do something on the off-chance, you do it because you hope that it will succeed, although you think that this is unlikely.
e.g. He had taken a flight to Paris on the off-chance that he might be able to meet Francesca.
他搭飞机去了巴黎，心想着或许能遇到弗朗西丝卡。
e.g. ...an American visitor who had turned up on the off-chance of catching a glimpse of the princess.
为有幸一睹王妃风采而前来的美国游客