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Say a player with a 16 passive perception is busy talking, fighting or even picking a lock, would they be able to passively perceive someone coming behind them or walking away from them?

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Passive perception is passive

By the rule, if you are not actively searching for something, then your passive perception is at play - if you stand guard, are on the lookout, searching for something, trying to observe subtle movement in the air or ground to discern where is an invisible creature, or trying to focus to hear some distant sound, then you do a perception skill check. Beside the times when you are explicitly using skill check you use (or rather, your GM uses) passive perception. If something is trying to sneak and it does not overcome your passive perception, then you hear, see, know it is sneaking, the same way enemies know if you are sneaking on them if you do not roll over their passive perception.

If the fact of being preoccupied with something would negate the use of passive perception, every single shop or tavern would be empty, because the "occupied" shopkeeper or barmaid would always fail checks to discern if someone is stealing from them when tending to the clients.

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    \$\begingroup\$ While I generally agree with this answer, I think it could benefit a lot from adding something about advantage/disadvantage on passive checks. I've seen many DMs granting a distracted creature disadvantage on passive perception. \$\endgroup\$ Commented Apr 11, 2022 at 12:42
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Cribbing from this answer:

Passive checks in general can be used whenever the DM wants to avoid dice rolls, either to speed things up (using the passive check as the average result over a period of time) or if the DM wants to determine a result secretly.

There are some special and specific rules for Passive Perception related to exploration and hiding. In the Movement section of chapter 8 of the PHB (Adventuring), there's this bit:

Noticing Threats

Use the passive Wisdom (Perception) scores of the characters to determine whether anyone in the group notices a hidden threat. [...]

Characters who turn their attention to other tasks as the group travels are not focused on watching for danger. These characters don’t contribute their passive Wisdom (Perception) scores to the group’s chance of noticing hidden threats.

This rule actually shows that despite the name "passive", it doesn't mean that this applies only when the character is passive ­— in fact, characters must be on the alert for danger to even get a check. It's just called "passive" because there is no active roll.

You would have to adjudicate whether the character is 'alert for danger', which is down to the GM to decide.

You could potentially rule that the distracted character has disadvantage, which subtracts 5 (as per the Basic Rules) from their passive check.

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