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From the Owlbear stat block:

Keen Sight and Smell: The owlbear has advantage on Wisdom (Perception) checks that rely on sight or smell.

Since the owlbear has a Passive Perception of 13 normally, does this mean that the owlbear has a Passive Perception of 18 for the purposes of seeing and smelling? Or does this only affect active checks? Does Passive Perception count as a check according to the rules?

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Passive Checks (page 162, Player's Handbook) says

A passive check is a special kind of ability check that doesn’t involve any die rolls. Such a check can represent the average result for a task done repeatedly, such as searching for secret doors over and over again, or can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster. Here’s how to determine a character’s total for a passive check: 10 + all modifiers that normally apply to the check

If the character has advantage on the check, add 5. For disadvantage, subtract 5. The game refers to a passive check total as a score. For example, if a 1st-level character has a Wisdom of 15 and proficiency in Perception, he or she has a passive Wisdom (Perception) score of 14. The rules on hiding in the “Dexterity” section below rely on passive checks, as do the exploration rules in chapter 8.

This rule is to allow the GM to be secretive. I would have no problem with a GM just rolling the perception check with advantage if he wants to, but by the rules the Owlbear in this example would have an 18 passive perception.

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Yes, advantage on a passive skill = +5, so in your example the Owlbear would have a passive perception of 18 for sight or smell. If he is actually making an active check, then you would roll with advantage as normal.

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