HELP! Should I rent to a tenant with a prior eviction?

[Garrett Diegel]

I have a unit in a tri-plex that is available to rent. I had a tenant call to see the property. Everything seemed good, they looked good and showed up on time to the showing. The only problem is when I sent them over the application they stated that they had a prior eviction 4 years ago in 2012. The tenant stated they would pay higher rent and a higher deposit but I'm concerned that I might end up with a problem on my hands in the event they decide to stop paying rent. On the plus side they were very open about the eviction and didn't try to hide anything. People can change so it's tough to just lable them. What would you do? Run the BG check and go from there?