The question calls for not only a bear defense gun but for a bear gun that doubles as a hunting gun. As most by now should realize, the hunting gun and the defense gun fill separate if related roles. The 12 ga shotgun loaded with slugs is suitable only for close range applications, although I am told that specialty shotguns can stretch their reach to medium range, but looses versatility for acquiring small game for camp food. Thus the smoothbore shotgun should be considered a less than ideal contender if big game hunting is conducted in places where longish shots are the rule. If you want to put game birds in the pot and have a bear defense gun, it remains a good choice.
Many bears have been killed with rifles from the .270 Winchester on up. But if you are thinking in terms of a protection specific rifle that you can also hunt with, I believe you need more. The versatility demanded by a bear defense/hunting rifle typifies a medium bore rifle with moderate to high velocity, heavy for caliber bullets from a light rifle with a relatively short barrel. Thus any big game animal can be taken out to 300 yards or more, yet the power is there to enthusiastically solve an emergency that occurs just a few feet distant. Contenders begin with the various .338s, .375s and .416s.
Large bore rifles in the .45/70 to .458 Winchester are superb up close problem solvers, but leave a bit to be desired if the anticipated range exceeds 200 yards, in that the shooter's range estimation becomes more critical than with flatter shooting cartridges. The .460 Weatherby loaded to it's potential is probably too much of a good thing for most, although it produces a flat trajectory when loaded with pointy bullets. A .458 Ultra wildcat would combine a trajectory suitable for shots out to 300 yards with less recoil than the Weatherby in a rifle of similar weight, particularly if 450 gr TSXs were chosen over 500s.
As to the likelihood of encountering a dangerous bear, or a bear that under certain circumstances becomes dangerous, that depends on the amount of exposure you have to bears in their environment. Thus my likelihood of having a run in with a dangerous bear exceeds the chances of someone who lives in an urban area or for that matter most rural areas. If your work involves bear control or research, your chances are again greater. If you travel through bear country on foot your chances of having a problem with a bear are greater than a person who spends an equal amount of time in the same area but travels by helicopter, in a vehicle, or on a quad. Consider though that I have lived in polar bear habitat for the last 20 some years. I have spent endless hours watching and walking around these guys on land and out on the sea ice. I've seen them on carcasses and I've seen them eat their young, but I have never seen a bear pull a seal up on an ice flow, kill it and eat it. Last spring a group of kids were up here for a 2 day visit and part of their visit involved going out in a Zodiac around the ice flows. Damned if they didn't see a bear pull a seal up on the ice and begin feeding! You just never know.
Bear defense is often dismissed as fantasy, but for the people who deal with bears regularly it is a serious issue. For the infrequent visitor to bear country, your chances of having a problem with a bear are low. What were the chances of those kids seeing a bear kill a seal? The chances were pretty low, but obviously not zero. Similarly your chances of having a dangerous bear encounter within the short time you are in bear country are low, but again not zero.