what does this phrase mean 我偏不信邪? the English literal translation is "I don't believe in evil" but that doesn't make sense to me.
偏: (6) [adv] wilfully; deliberately; persistently
不信 = don't believe
邪: (2) disaster; misfortune (caused by evil spirits)
I, despite everything, do not believe there are supernatural coincidences
I insist, I do not believe there are supernatural coincidences
For example: You lost seven coin tosses in a row by choosing head, and for the eighth time, you still insist on choosing head, because having tail win eight times in a row is not scientific, the only way it could happen is there's some evil spirits power at work. And you don't believe there are evil spirits.
Well, if I lost seven coin tosses in a row, the most logical explanation I can think of is -- the other guy is cheating.
Context?? What is the context??
不信邪：refuse to be taken in by fallacies; not dread evil forces
Depending on the context, probably just means, "I'm not inclined to believe you/that."
Or think of Ghostbusters: I ain't afraid of no ghost!
Edit: Depends on the context of course, but most likely the meaning is:
I don't believe that I can't do it.
偏 is 就, referring to one's biased determination. 我偏不信邪 = 我就不信邪.
偏不信 can be simply/blindly refused to believe.
邪 is used in the sense as in 邪门, indicating something goes unexpected, abnormal, odd, illogical, etc..
我偏不信邪 means I simply refuse to believe in the way how it goes (because it's been very odd, abnormal, illogical and unexpected).
我偏不信邪 also implies that I will go the other way around.
It generally means that when something (usually a bad thing) happened or failed frequently, some people believe unshakeably that it's some abnormal or strange thing, it must be controlled by god, ghost, demon, or there is some supernatural power in the background. But if I think that it's just a coincidence, or that I am not hard nearly enough, I am sure that I am able to find out the truth or succeed finally, I can say 我偏不信邪, 我就不信这个邪, means that I don't believe that heretical idea.
Similar to Tang Ho's answer, think of it like someone playing poker and going on a losing run. They might justify continuing with something like "I don't believe in bad luck" or "I refuse to believe in bad luck". 邪 doesn't have to involve literal ghosts or spirits or evil any more than 什么鬼 has to involve literal ghosts--it can be an allusion to superstition.