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Having a highly intelligent familiar in D&D 5e, I noticed that sprites have higher intelligence and wisdom than my character, and obviously can do a better job on analysis or insight rolls. Judging by his perception bonus, I figure he has +2 proficiency bonus, as well as double proficiency bonus to his stealth skill.

What other skills can a sprite familiar use and in which is he proficient?

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What A Familiar Can Do

The sprite familiar can do anything a Sprite enemy/NPC can do acording to their stat block. If a familiar has something like Blind-sense (bat) or better perception than you, you can ask/tell it to look around for you.

The stats block only list bonuses that are proficient, other scores (like Arcana or Religion [+2]) would be determined by the typical way that is done for a player. The Monster Manual lists the mods after the stat, and you'll see they follow the same rules as players (+1 to modifier for every 2 points after over 10, -1 for every 2 points under 2). With the sprite, that means s/he could do a athletics check, but his/her modifier is -4. Yes, you're reading the stats right, are extra bonuses.

Strenght Save:  -4
Athletics:      -4
Dex Save:       +4
Acrobatics:     +4
Slight of Hand: +4
Stealth:        +8
Con Save:        0
Int Save:       +2
Arcana:         +2
History:        +2
Investigation:  +2
Nature:         +2
Religion:       +2
Wis Save:       +1
Animal Handing: +1
Insight:        +1
Medicine:       +1
Perception:     +3
Survival:       +1
Cha Save:        0
Deception:       0
Intimidation:    0
Performance:     0
Persuasion:      0

Passive Perception: 13

It can, even, perform the "help" action for you or a party member for any action the Sprite is able to do. It might even make sense to help your sprite look around (i.e. Advantage with the higher perception modification). Likewise it is terrible at lifting (athletics), but it can still "help" you lift providing you advantage with your strength.

The familiar can provide advantage on attacks to an enemy if he/she is within 5ft, and spends its action "helping" by distracting the enemy (which it can do while invisible) or by flanking rules (which it has to be visible to do).

The familiar can also be dismissed and brought back in a different form (bat, cat, etc.) following the rules in the Find Familiar Spell.

The familiar can, of course, deliver touch spells cast by the Warlock.

What A Familiar Can't Do

The stat block doesn't list any tool proficiency, so the sprite can't pick locks or like. Which means that it can't provide "help" for those tasks either.

As a Warlock, your sprite can attack but at the cost of an action, but unless you can't act for some reason, an Eldritch Blast or other spell is going to be more effective.

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From a rule based perspective, don't assume Proficiency in a Skill unless it is listed. Otherwise, a Sprite can try to use any Skill and be treated as a player would be treated.

For a skill attempt roll your Warlock makes, having the Familiar Help (which yields Advantage, Basic Rules p. 72 Help) is probably the way to go unless you can't work together. Bonus gained by Advantage is roughly equivalent to "+5" on your roll. (Example: Passive Check Basic Rules p. 58).

From your example, doing your own Insight roll with a +5 bonus (Sprite Helps) will likely beat the Sprite's Insight check with a +2 bonus from its Intelligence. (Basic Rules p. 59, Working Together).

Proficiency Bonus
Characters have a proficiency bonus determined by level. Monsters also have this bonus, which is incorporated in their stat blocks. The bonus is used in the rules on ability checks, saving throws, and attack rolls.

Basic Rules (p. 58) match skills and ability scores.(Stats) The formula for monsters looks like 2xProficiency + attribute bonus for some listed skills, where they have a special expertise, and Proficiency + attribute bonus for others listed.

  1. Sprite as the first example.

    The Perception Skill is listed as +3, CR is 1/4 for a Proficiency bonus of +2. (Wisdom 13/+1) 2+1 = 3, and for passive Perception 13.

    The Stealth Skill is listed as +8, but the Dex (18) bonus of +4 plus +2 proficiency gives me 6. In this case, it the proficiency bonus has been doubled to yield (4+2+2) 8, most likely due to its expertise in hiding. This is similar to the "Expertise" feature of the Rogue earned at 6th level. (P. 27 Basic Rules).

    Expertise

    At 1st level, choose two of your skill proficiencies, or one of your skill proficiencies and your proficiency with thieves’ tools. Your proficiency bonus is doubled for any ability check you make that uses either of the chosen proficiencies.

    Your Sprite's passive Insight score would be 10 plus 2 (Int 14/+2) = 12 since the Sprite does not have skill listed in its description and would not benefit from Proficiency. (Your DM could rule otherwise at a given table). Treat all other skill checks as you would for a player.

  2. Aboleth is another illustration of special skills/expertise. At CR 10, it's proficiency bonus is +4, and it appears to have "expertise" in two areas.

    Skills History +12, Perception +10

    If you double the +4 Proficiency bonus and apply the Int bonus (18/+4), you get a 12.

    If you double the +4 Proficiency bonus and apply Wisdom bonus (15/+2) you get 10.

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