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You know what? I change my mind. I read the mod's guidance on this, and It's clear that Bug Out Bags are decidedly for /out/door situations, and use of a pellet gun would certainly fall into the same category.
So, OPAnon, I will answer your question:
>Why would you buy a pellet gun for a BOB / SHTF situation?
The answer is this: Because pellet rifles can take small game, and if you're smart enough to get a pneumatic (like pic related) as long as you have pellets and an arm to pump the rifle, you have the ability to gather meat.
The .22LR Rifle at top would be a superior choice, owing to it's ability to gather larger game at farther distances as well as serve as a self-defense weapon, but that comes at the cost of weight and less ammunition.
Personally, that's a good trade for me, but if you live in a country where you'll get nicked by Old Bill who'll do you up on a weapons charge ping pong piddley bobs your uncle apples and pears knickers spanner lift bonnet, you may be better off with the air rifle.