Is it possible for a staff member to do a better job they weren't trained for in comparison to a staff member who was trained thoroughly? Why?

Yes. But it will take time and he/she will learn through hit and trial method. He/she can compete trained staff but with struggle hard work and adaptability. People are not same. they have different skill sets, compassion level, talents and hard work level and smart work levels. 

Yes, but very rare

it depends on the training subject , The experience , the personality

some people can adapt faster , and some people have great attitude and self directed learning which is most powerful style to get ahead, and they mostly leaders by nature also.

Yes of course.

Because people are different.  They have different experiences, talents and characters.

To be fair, if somebody has been "trained thoroughly" and is performing according to their training, then it is likely that performance in the basic tasks of the job cannot be surpassed because it is already excellent.  However, really good people will find ways of doing things better than the training.

No, It isn't possible for new member to perform well as compare to well trained staff member. Training is very essential tool to groom and enhance the efficiency of any individual with confidence.

It might happen but it is very unusal. 

it depence from the staff's attitude and from the personal skills. Some people are naturally inclined in some aereas than in other or same can learn faster the others.

Yes it is ,because for person to do their job right he needs to recognize need for him to be more involved , and to anticipate . For us to do our job ,it is more then us being trained . We need to be enthusiastic hard working and willing to learn more every day.

Thank you for the invitation.

Of course employee who was trained to do his job better, because employee training polishing skills to lead his work efficiently and effectively.

Its impossible because it will take a training of staffs to develop the employee's competencies, and they will know the very hazards associated with the activities and the precautions with control measures that have to be in place prior to the start, also know  their limits in the tasks in addition with what they ought to do in the case of any emergency of which will reduce the level of supervision on the tasks. 

Well, if we exclude exceptional cases...

A trained employee will do a better job than untrained employee.

So there is no question to believe that untrained employee will trained one.

For example: - It’s not possible for novice to work on computer (like excel, etc.)  And generate report until and unless he or she is trained.   

Training on new skills, competencies is important to meet changing needs. Very important for increasing productivity and boosting employee morale and career development

Training is very essential part of all job functions, it helps to strengthen skills and competencies. Hence, a trained staff can perform a job more efficiently than untrained staff.