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Unfortunately, employment in the state of Nevada is "at will" absent an express agreement to the contrary. At will employment can be terminated for any reason not amounting to discrimination on the basis of a legally protected trait (race, religion, gender, etc.) or retaliation for engaging in certain forms of legally protected conduct (filing a wage claim, taking FMLA leave, etc.). It doesn't matter whether the basis for termination is fair, reasonable or even true. It also doesn't matter whether the employee is "probationary" or full time.
So, regrettably, there generally is nothing that you could typically do about an employer rescinding a full time offer based on information contained in a background check. The only exception would be if the information that they based their rescission on was a prior wage claim, workers comp claim, sexual harassment lawsuit, or some other conduct that might be considered legally "protected." If they based the rescission on a criminal conviction, poor reference from a former employer, or most of the other things employers look at when conducting background checks, there will be no violation of the law and, as an at will employee, any such discoveries can lawfully be used as a basis to deny employment.
The silver lining to all of this is that you should have no problem getting approved for unemployment benefits as you look for a new job.
I hope that you find this information helpful and am genuinely sorry if it is not what you were hoping to hear. Please do not hesitate to let me know if you have any questions or concerns regarding the above and I will be more than happy to assist you further.
If you do not require any further assistance, please be so kind as to provide a positive rating of my service so that I may receive credit for assisting you. Very best wishes moving forward.