Character Alice wants to pickpocket Character Bob. If Bob isn't actively anticipating this (e.g., sitting & drinking some ale), does Bob roll a perception check or does Bob use his passive perception score?
If Bob is actively anticipating the pickpocketing, does Bob get advantage on the perception roll when Alice attempts to pickpocket him?
Passive checks in 5e aren't about characters "doing something passively". They are about players not making rolls (that's why they are "passive"), but DM still getting a check result:
A passive check is a special kind of abilily 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 (Player's Handbook, page 175)
When Character A wants to pickpocket Character B, the player should make a roll, or DM can use a passive check instead. The decision is up to DM, it depends on if the DM doesn't want the player to suspect of something by rolling a die.
DM can also use passive checks when the result should be more predictable - for instance, Perception check for a guard on duty. In that case DM can make it for an NPC. But the main use case is making a passive check for a player's character without warning the player.
As a DM, I would separate this into two separate actions. Alice needs to (1) get close enough to Bob to attempt to pickpocket him, and then (2) actually pick his pocket.
How Alice handles getting close to her target is situation-dependent, and there are several different options. She could find a reason to walk into Bob, or she could try and unobtrusively walk behind him, or she could sit next to him and start a conversation. You could resolve these with (for example) Deception, Stealth (versus passive perception), or Persuasion, with success indicating that Bob isn't suspicious of Alice's proximity. There are many other possibilities as well.
Once Alice is close enough to Bob, she can attempt to actually pickpocket him; this is a Sleight of Hand check at whatever DC you as the DM determines is appropriate. DC15 is a good default, and you might adjust it based on how successful Alice was at getting close to Bob. For example, if she sat next to Bob and started a conversation, but failed her Persuasion check, you might narrate that Bob was suspicious and scooted his chair away a bit, raising the DC to 20.
Since Sleight of Hand is already inherently stealthy, I would rule that any success results in undetected pickpocketing, and a failure doesn't result in Alice's pickpocketing being detected. (If Alice rolls a 1, I might narrate that they bungled the attempt so badly that it was obvious what they were doing).
If Bob is being vigilant — intentionally watching for danger — then I would let him make an active Perception check to notice a pickpocketing attempt.
I don't think this exact situation is covered in the rules. There are many ways to make a fair ruling at the table. Here is a list of a few examples, but I am sure there are many more ways to rule this situation.
Personally, I prefer the last alternative as it has a chance of some interesting outcomes that are impossible in the other alternatives.
As the PHB says under Passive Checks (p 175):
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.
I've highlighted the two reasons given for using a passive check, and expand on each below.
1. The average result for a task done repeatedly
Firstly, doing things repeatedly: passive Perception is normally applied to guards doing their job, which is constantly keeping watch, and to adventuring PCs who aren't distracting themselves with other tasks such as making maps. It represents a state of alertness in which a creature is on the look out for trouble.
Your character B drinking ale is not on the look out for trouble, but their senses haven't gone completely AWOL either: they should have a chance of spotting trouble, but not as much as if they'd been guarding a gate or crawling through a dangerous dungeon. For creatures like this, it's reasonable to allow passive Perception, but at disadvantage to represent the fact that they are not being alert.
If character B were making a map, or searching for treasure in foot locker, then they are distracted. The rules give the DM latitude to rule that such a character cannot benefit from passive Perception. But to be honest, character B is probably about as alert to danger as they would be nursing a pint in a tavern. So again, a passive Perception check with disadvantage is justified.
2. When the DM wants to secretly determine the outcome
A common reason for secretly determining the outcome of an ability check is to reflect ignorance or a lack of awareness. The passive check keeps both the PC and is player in the dark, and it avoids meta-gaming. That's appropriate here: character B is unaware of character A's intentions and so can't play an active part in the scene, unless and until they notice what's going on. And here noticing is a purely passive thing.
So a passive Perception check - with disadvantage - is warranted by the rules on passive checks.
