Achieving highest Rogue DPR at 3rd level using PHB+1

Question

I think this is a straight-forward optimization question. The restrictions are:

• Assume enemies have 16~18 AC ¹. Saving Throws have +5 bonus.
• Rogue is ranged (i.e. it has to fight from a decent range, preferably larger than 30 ft), 3rd level.
• Book material is restrained to PHB+1. (This is not an AL character; we're just using the PHB+1 rule.)
• The highest rolled stats are 17 and 15, which can be put anywhere (altho I'm guessing the 17 is going to Dex). Not sure Rogue needs any secondary stat for DPR, but the info is there.
• Rogue has access to one uncommon permanent magic item.
• Assume there's place for hiding every turn, if that's the best use for Bonus Action using Cunning Action.
• One level can be dipped, making it a Rogue 2/Something 1. The first level is rogue, though.
• For Sneak Attack purposes, the Rogue has melee allies to trigger it.
• Any mundane item is fine, as long as it has a reasonable GP cost for a 3rd level character, so roughly 300 gp (for the entire build).
• As a note, this is an actual character I intend to play. Thus, if you are going to suggest something really crazy ("buy a ballista and fire with it"), please explain how I'm supposed to enter a dungeon with that. (That's to say: it's not just a random thought experiment, it needs to be actually playable in a real table).
• The DPR should be sustained (i.e. not Nova/Bursts), but doesn't need to be infinite, sustaining it for ~5 rounds is enough.

I need some help because

• I'm not used to optimization choices out of the PHB (I've been mostly DM'ing and I'm not exactly worried about optimizing PCs while doing so) and it seems SCAG has presented some good Rogue subclasses.
• Enemies' AC is higher than usual for this level, so I'm not sure my standard Variant Human with Sharpshooter will be good - even if it is, I'm interested in knowing if there are better options.
• The multiclassing Rogue 2/Something 1 also makes it a little more challenging, as +2 on hit from Archery fighting style (against kinda high AC enemies) seems good, but not sure it's worth the loss of 1d6 in the Sneak Attack, but I can only do the math after I have sorted everything else (like "Am I still using sharpshooter?")

And the objective is, as the title says: maximize the DPR (Damage per round) of the Rogue.

Note: I would rather having no shenanigans². Before anyone asks, yes, I'm aware Rogue is not the best choice for DPR, mainly this early.

¹ If it happens that the build with max DPR changes during this small range, the best answer is the one that includes both builds, explaining in which situation each one should be used. Many (most) optimization questions have answers stating the best build from 0 to 20 AC, so I don't see this as a problem.

² For the purposes of the question, shenanigan is anything that a DM is most likely going to tell you "Nope", anything that is not how the rules actually work or anything that is stretching RAW too much, clearly going against intention (although I'm not sure how common this is in 5e). Other than that, shenanigans are usually DM adjucation, so if the answer is given in good faith (i.e., not clearly trying to push an exploit) I'm fine with it.

Answer 1

So, I couldn't sleep and decided to work on that. Here's the best I got until the moment - using only PHB.

I have done the math for 3 builds so far, all using Variant Human and fighting style Archery when multiclassed:

• Rogue 3, Crossbow Expert, Hand Crossbow +1;
• Rogue 2/Fighter 1, Sharpshooter, Heavy Crossbow +1;
• Rogue 2/Fighter 1, Crossbow Expert, Hand Crossbow +1;

Sneak Attack procs are assumed every round, as there are melee allies. I'm the first to say I was not very creative with the use of subclasses or magic items. As noted by András in the comments, Sneak Attack only procs once per turn, so the second Hand Crossbow attack does not deal the extra 2d6 damage, putting both builds roughly 2.5 average points below the Sharpshooter.

Assuming you you can hide to get unseen every turn

If Hide = Unseen = Attack with advantage is a thing (this depends on your stealth checks, but with expertise they should be very likely to work), the highest damaging build becomes Rogue 2/Fighter 1, Sharpshooter, Heavy Crossbow +1.

1. Cunning action for hiding.
2. Attack with advantage. Damage on hit = \$5.5 (1\textrm{d}10) + 3.5 (1\textrm{d}6) + 10 + 5 = 24\$. Damage on critical = \$11 + 7 + 10 + 5 = 33\$.
3. The probability of hitting without a crit is given by \$1 - (1 - \textrm{Usual probability})^2\$, where Usual probability is calculated using the +4 Modifier. Against ACs 16, 17, and 18, that works out to \$\{0.64, 0.5775, 0.51 \}\$ respectively.
4. The probability of a crit with advantage is \$1 - 0.95^2 = 0.0975\$.
5. The final average damage is \$ \{18.57, 17.07, 15.45 \} \$, for ACs 16, 17 and 18, respectively.

---

For the Rogue 3, Crossbow Expert build, assuming we can attack with the same hand crossbow twice, we have nine possible situations. Each one has a \$P_i\$ associated with it and an average damage \$ d_i \$. The average total damage is given by \$ \sum_{i=1}^{9} P_i \cdot d_i \$. Let \$ P = \{ 0.55, 0.5, 0.45 \} \$ be the probability of hitting an individual hit and \$ P_c = 0.05\$ the probability of landing a critical hit.

1. First attack hits, second misses. \$ P_1 = P \cdot (1 - P - P_c) \$; \$ d_1 = 3.5 + 7 + 5 = 15.5\$.
2. First attack misses, second hits. \$ P_2 = (1 - P - P_c) \cdot P \$; \$d_2 = 15.5\$.
3. First attack crits, second misses. \$ P_3 = P_c \cdot (1 - P - P_c) \$; \$d_3 = 7 + 14 + 5 = 26 \$.
4. First attack misses, second crits. \$ P_4 = (1 - P - P_c) \cdot P_c \$; \$d_4 = 26 \$.
5. First attack hits, second crits. \$ P_5 = P \cdot P_c \$; \$d_5 = 3.5 + 7 + 5 + 7 + 5 = 27.5 \$.
6. First attack crits, second hits. \$ P_6 = P_c \cdot P \$; \$d_6 = 7 + 14 + 5 + 3.5 + 5 = 34.5 \$.
7. Both attacks crit. \$P_7 = P_c \cdot P_c\$; \$d_7 = 7 + 14 + 5 + 7 + 5 = 38 \$.
8. Both attacks hit. \$ P_8 = P \cdot P \$; \$ d_8 = 3.5 + 7 + 5 + 3.5 + 5 = 24\$.
9. Both attacks miss. \$P_9 = (1 - P - P_c) \cdot (1 - P - P_c)\$; \$d_9 = 0\$.

The result of the summation, if I did everything right, gives us an average damage of \$ \{16.92, 15.79, 14.62 \} \$.

Similarly, for Rogue 2/Fighter 1, removing the 1d6 from Sneak Attack, we get \$\mathbf{d} = \{12, 12, 19, 19, 24, 27.5, 31, 20.5\} \$, but increasing the Probabilities to \$ \{ 0.65, 0.6, 0.55\}\$, we get an average damage of \$ \{ 15.66, 14.70, 13.73 \} \$.