The phrase "run luck" refers to relying on chance or good fortune to achieve success. As with most idioms, there are several synonyms that can be used to convey a similar meaning. Some options include "bank on serendipity," "hope for a stroke of luck," "gamble on fate," or "depend on the fates." Other phrases that could be used include "taking a chance," "rolling the dice," or "throwing caution to the wind." No matter which phrase you choose, they all convey the sense of relying on luck or chance to achieve a desired outcome, rather than through careful planning or effort.