In line with this great question, what's the possibility of a commercial airplane crashing due to a computer error? Or are there already known cases of this? As a passenger, should you be worried about this possibility happening to you someday in the future?
First; let's be honest. No one really knows what the exact probability is.
But we do have a design certification system that requires developers to apply various development assurance processes to meet specific safety standards for the design. The primary requirement is covered by §25.1309 Equipment, systems, and installations.
That has a lot of words about identifying potential faults and mitigating the associated hazards. But there are no numbers. As with most rules, the FAA conveniently provides an Advisory Circular to help you show compliance with the rule. In this case, its 25.1309-1A - System Design and Analysis.
The acceptable standard for critical faults (those with a potential catastrophic result) is found at the bottom of para 10.
(3) Extremely Improbable failure conditions are those having a probability on the order of 1 X 10^-9 or less.
So, if the developers do their job to the standard and the standard (and the methods used to evaluate it) are accurate the answer to the question is:
The probability is less than once in every 1 billion hours of operation.
As a point of statistics, that number applies to the cumulative hours of all aircraft of that design.