Final answer:
The most likely nerve to be affected by a mid-shaft humerus fracture after falling on an outstretched hand is the radial nerve due to its proximity to the fracture site.
Step-by-step explanation:
The patient with a fracture of the mid-shaft of the humerus after falling on an outstretched hand is most likely to have an injury to the radial nerve. This nerve runs in close proximity to the mid-shaft of the humerus, and fractures in this region can result in nerve damage. The other options listed - axillary, median, and posterior interosseous nerves - are more likely to be affected by other types of injuries or fractures in different locations.