[RPG] Do you add Proficiency bonus AND the stat modifier to skill rolls

Passive perception is exactly that, passive. It's what the PCs are always using when not actively searching for something and doesn't use a roll of the die.

To determine if you should use passive perception or allow a player to roll, listen to what they say their PCs are doing. If they say they are standing watch, keeping an eye out or something similar, they are actively searching so they can make a wisdom (perception) check, otherwise they are using their passive perception.

Although, it's kind of an experience call as the DM and depends on what the PCs are actually doing and whether or not the DM gives them the benefit of the doubt. In the case of the video, one guy said he was keeping a look out, thus actively searching and two of them were scouting off to the side of the road so the DM ruled they were also searching, rather than just wandering off into the bushes for no real reason, and allowed those three PCs an active wisdom (perception) check, whilst the others would be using their passive perception.

Skill modifier = relevant ability modifier + proficiency bonus (if proficient) + other modifiers

What follows is a distillation of much of Chapter 7 of the PHB, "Using Ability Scores"

• relevant ability modifier: each skill usage is associated with an ability; you can find which either in tiny letters next to the skill on a WotC character sheet or in a chart on PHB p.174.^* Your skill modifier has, as one component, that ability’s modifier.

• proficiency bonus (if applicable): if you are proficient in the skill, then you also add your proficiency modifier to the skill modifier. You obtain skill proficiencies from your class, from your background, and possibly from downtime activities.

• other modifiers: you might pick up some other modification, the most commonly Rogues' "expertise" or Bards' "Jack of all Trades" class features. Their impact on skill modifiers is described in their respective class descriptions.

An example:

Stealth: +4 modifier

• your DEX modifier is +2

• you are proficient and (assuming you're starting this character at level 1) have a proficiency bonus of +2

• you have not mentioned being a rogue with expertise applied to Stealth, so I'll assume no other modifiers. (Jack of all Trades kicks in at level 2, and I'm still assuming level 1)

^{* - Okay, it's actually a bit more than that. Skills aren't uniquely tied to the ability listed on the sheet or in that chart; those are simply the most common abilities that invoke those skills. (And for many GMs they're the only ones that ever do.) If you read chapter 7 carefully you'll see that what we're describing aren't skill checks, but are ability checks to which a skill proficiency might be applied. So when walking into the Wizengamot and trying to talk circles around the gathering, a GM might find it appropriate to ask for an Intelligence-based intimidation check. See also the DMG p.239 or Angry GM's screed on the topic.}