Criminals wont think about how bad prison is because they don't have enough sense to think about how bad prison can be but if they know that they could die because of their crimes, they might just reconsider.
Sure, death penalty is really efficient at preventing crimes.
Proof is that USA definitely has low crime rates! xD
The reason for the crime may be anything regardless of the country. And fear of deah penalty might not even be a holdback for the crime but at least we would be sure that this person won't commit crime again and the family of the victim would get satisfaction with justice.
And this is why people don't avenge in modern society, because we have judicial system for this.