Short answer: It’s a rule of thumb to help guage how to react with a scary predator 80% of the time. Unfortunately, it’s not that simple. You should act based on how the bear is behaving (defensive vs. curious vs. predatory) more than its color.
Long answer: The rule of thumb is due to typical bear behavior and evolution.
Black bear habitat is heavily wooden areas where they evolved to run away or climb up trees to escape threats rather than fight. They are also comparatively smaller and more easily intimidated.
Brown bear habitat is typically more wide open areas where retreat is more difficult. They are larger and more commonly resolve conflicts by stand offs and eliminating the threat. If you tried to stand off with a brown bear, you will likely lose. By pretending to be dead, you show the threat has been eliminated already, and the bear doesn’t need to keep attacking. Importantly, if the bear starts trying to eat you at this point, you SHOULD fight back.
Polar bears evolved in expansive wide open areas where any source of food is few and far between. They are massive and skilled predators. Few people have encountered a polar bear and survived without a weapon, vehicle, or structure. If a polar bear has found you and decides it wants to eat you, your best bet is to hope it ends your life quickly rather than be eaten alive. [There was one person who survived an encounter by stripping their clothes,](https://nypost.com/2020/08/29/get-naked-how-to-fend-off-a-polar-bear-attack/) but that doesn’t rhyme as well.
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