Yes, you can refuse to rent to a prospective tenant if they are on probation or have a criminal past, so long as you are not discriminating against someone who is within a protected class.
Frank W. Chen has been licensed to practice law in California since 1988. The information presented here is general in nature and is not intended, nor should be construed, as legal advice for a particular case. This posting does not create any attorney-client relationship with the author. For specific advice about your particular situation, consult with your own attorney.
You can generally discriminate for anything that isn't a protected status, like sex, age, religion. There is no law that I am aware of that protects anyone from being discriminated against for having been incarcerated. However, you might want to check references before you flat out refuse to rent to someone on this basis. Many people have made mistakes, but have turned their lives around.
I agree with my colleague that as long as the basis for the decision is not a person's protected status such as age, gender, religion, disability, etc, it is allowable to discriminate.
I have been licensed to practice in the State of Oregon since 1990. I am not offering legal advice regarding your question, only general information regarding the law. You are not my client nor am I your attorney unless we sign a retainer agreement.