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Monoclass DPR Comparison: Eldritch Knight Archer vs Melee Arcane Trickster
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<blockquote data-quote="Esker" data-source="post: 7628412" data-attributes="member: 6966824"><p>If it is low, it means a smaller share of attack rolls have the opportunity to benefit from precision attack than in your calculations, so the average damage will be reduced.</p><p></p><p></p><p></p><p>Yes, this is a good point: when there are fewer than average rolls in the precision attack range, those have to go somewhere, either to automatic hits or to misses where we are not using our dice. So we should consider whether the cases where we have fewer-than-expected triggers for precision attack look different than the cases where we have the expected number or more.</p><p></p><p>In the ideal case, in a batch of 20 rolls, we have 20%, or 4, in the precision attack range, each of which has an 81.25% to become a hit, and 16 outside of that range. In the case of AC 16 (where the chance to hit if you can't use precision dice is 40%), the 16 that fall outside the precision attack range have a 50% chance of being hits. So, our overall chance to hit is 4/20 * 0.8125 + 16/20 * 0.50 = 0.5625. So, precision attack is giving us an extra 16.25% to our to-hit chance, or an extra 3.25 hits out of 20, as we saw before.</p><p></p><p>But what about the times when we only get 3 rolls in the mid-range? The missing one has exactly the same distribution of possible values as the 16 we assumed didn't trigger precision attack before, so we get a to-hit chance of 3/20 * 0.8125 + 17/20 * 0.50 = 54.6875%. With 2 in the mid-range, it's 53.125%, and with 1 it's 51.5625. Of course, if there are more rolls in the "within 4" range than we have precision dice, those are automatic misses. Overall, the chance to hit over N rolls between short rests is given by the formula</p><p></p><p>[sum for n = 0 to n = N] P(n rolls within 4) * [min(n, 4) * 0.8125 + (N - n) * 0.50 + max(n-4, 0) * 0.00]</p><p></p><p>where n is the number of rolls that meet our criterion for triggering precision attack.</p><p></p><p>P(n rolls within 4) is given by the binomial distribution over N rolls with a "success" probability of 0.20. </p><p></p><p>With some help from R, this is easy to compute:</p><p></p><p><span style="font-family: 'courier new'">AC <- 16 # enemy AC</span></p><p><span style="font-family: 'courier new'">atk<- 3 # attack bonus after -5 penalty </span></p><p><span style="font-family: 'courier new'">m <- AC - atk # smallest natural roll to hit w/o precision</span></p><p><span style="font-family: 'courier new'">N <- 20 # Number of attack rolls per short rest</span></p><p><span style="font-family: 'courier new'">t <- 4 # Maximum distance when we will use a die</span></p><p><span style="font-family: 'courier new'">n <- 0:N # Number of attack rolls within t of a hit</span></p><p><span style="font-family: 'courier new'"></span></p><p><span style="font-family: 'courier new'"># Chance of n rolls of N within t</span></p><p><span style="font-family: 'courier new'">p_n <- dbinom(n, size = N, prob = t / 20) </span></p><p><span style="font-family: 'courier new'"># Chance to hit when using precision attack</span></p><p><span style="font-family: 'courier new'">p_prec <- sum(1/t * (1/8) * (8-t+1):8)</span></p><p><span style="font-family: 'courier new'"># Chance to hit when not triggering precision attack</span></p><p><span style="font-family: 'courier new'">p_other <- (20-m+1) / (20 - t)</span></p><p><span style="font-family: 'courier new'"></span></p><p><span style="font-family: 'courier new'">to_hit <- sum(p_n * (pmin(n,4)/N * p_prec + (N-n)/N * p_other))</span></p><p><span style="font-family: 'courier new'">to_hit</span></p><p><span style="font-family: 'courier new'">[1] 0.5341341</span></p><p><span style="font-family: 'courier new'"></span></p><p><span style="font-family: 'courier new'"><span style="font-family: 'arial'">Actually, this didn't take action surge into account: we're assuming 23 attack rolls, not 20. So, changing N to 23, we get</span></span></p><p><span style="font-family: 'courier new'"><span style="font-family: 'arial'"></span></span></p><p><span style="font-family: 'courier new'"><span style="font-family: 'arial'"><span style="font-family: 'courier new'">to_hit</span></span></span></p><p><span style="font-family: 'courier new'">[1] 0.524508</span></p><p><span style="font-family: 'courier new'"></span></p><p>Slightly lower to-hit, but of course more attacks. With the 16.5 damage per attack, that's</p><p></p><p><span style="font-family: 'courier new'">dph <- 16.5</span></p><p><span style="font-family: 'courier new'">daily_attacks <- 3 * N</span></p><p><span style="font-family: 'courier new'">total_damage <- daily_attacks * to_hit * dph</span></p><p><span style="font-family: 'courier new'">total_damage</span></p><p><span style="font-family: 'courier new'">[1] 597.1524</span></p><p><span style="font-family: 'courier new'">DPR <- total_damage / 20</span></p><p><span style="font-family: 'courier new'">[1] 29.85762</span></p><p><span style="font-family: 'courier new'"></span></p><p>So, like I said, about 1.5 below what we'd get if we had exactly the "right" number of uses of precision attack every rest. For other ACs, here's what the table looks like with 7 rounds per short rest.</p><p></p><p><span style="font-family: 'courier new'"> total_per_sr DPR</span></p><p><span style="font-family: 'courier new'">11 305.4 43.6</span></p><p><span style="font-family: 'courier new'">12 285.6 40.8</span></p><p><span style="font-family: 'courier new'">13 265.8 38.0</span></p><p><span style="font-family: 'courier new'">14 246.0 35.1</span></p><p><span style="font-family: 'courier new'">15 226.2 32.3</span></p><p><span style="font-family: 'courier new'">16 206.4 29.5</span></p><p><span style="font-family: 'courier new'">17 186.6 26.7</span></p><p><span style="font-family: 'courier new'">18 166.8 23.8</span></p><p><span style="font-family: 'courier new'">19 147.0 21.0</span></p><p><span style="font-family: 'courier new'">20 127.2 18.2</span></p><p></p><p> </p><p></p><p>I'm not sure what you're saying here... Precision attack is responsible for more than half of the fighter's edge over the rogue, which we now know how to quantify (thanks, by the way, for spurring me to think about this!). Without superiority dice, the fighter's DPR (vs AC 16) over 20 rounds, assuming three action surges, is just</p><p></p><p>23 * 3 * 0.40 * 16.5 / 20 = 22.8</p><p></p><p>which is only about 27% better than the rogue if we assume the rogue gets advantage 75% of the time. (The fighter will get advantage from party members some of the time too, so it's a bigger edge than that, but y'know)</p></blockquote><p></p>
[QUOTE="Esker, post: 7628412, member: 6966824"] If it is low, it means a smaller share of attack rolls have the opportunity to benefit from precision attack than in your calculations, so the average damage will be reduced. Yes, this is a good point: when there are fewer than average rolls in the precision attack range, those have to go somewhere, either to automatic hits or to misses where we are not using our dice. So we should consider whether the cases where we have fewer-than-expected triggers for precision attack look different than the cases where we have the expected number or more. In the ideal case, in a batch of 20 rolls, we have 20%, or 4, in the precision attack range, each of which has an 81.25% to become a hit, and 16 outside of that range. In the case of AC 16 (where the chance to hit if you can't use precision dice is 40%), the 16 that fall outside the precision attack range have a 50% chance of being hits. So, our overall chance to hit is 4/20 * 0.8125 + 16/20 * 0.50 = 0.5625. So, precision attack is giving us an extra 16.25% to our to-hit chance, or an extra 3.25 hits out of 20, as we saw before. But what about the times when we only get 3 rolls in the mid-range? The missing one has exactly the same distribution of possible values as the 16 we assumed didn't trigger precision attack before, so we get a to-hit chance of 3/20 * 0.8125 + 17/20 * 0.50 = 54.6875%. With 2 in the mid-range, it's 53.125%, and with 1 it's 51.5625. Of course, if there are more rolls in the "within 4" range than we have precision dice, those are automatic misses. Overall, the chance to hit over N rolls between short rests is given by the formula [sum for n = 0 to n = N] P(n rolls within 4) * [min(n, 4) * 0.8125 + (N - n) * 0.50 + max(n-4, 0) * 0.00] where n is the number of rolls that meet our criterion for triggering precision attack. P(n rolls within 4) is given by the binomial distribution over N rolls with a "success" probability of 0.20. With some help from R, this is easy to compute: [FONT=courier new]AC <- 16 # enemy AC atk<- 3 # attack bonus after -5 penalty m <- AC - atk # smallest natural roll to hit w/o precision N <- 20 # Number of attack rolls per short rest t <- 4 # Maximum distance when we will use a die n <- 0:N # Number of attack rolls within t of a hit # Chance of n rolls of N within t p_n <- dbinom(n, size = N, prob = t / 20) # Chance to hit when using precision attack p_prec <- sum(1/t * (1/8) * (8-t+1):8) # Chance to hit when not triggering precision attack p_other <- (20-m+1) / (20 - t) to_hit <- sum(p_n * (pmin(n,4)/N * p_prec + (N-n)/N * p_other)) to_hit [1] 0.5341341 [FONT=arial]Actually, this didn't take action surge into account: we're assuming 23 attack rolls, not 20. So, changing N to 23, we get [FONT=courier new]to_hit[/FONT][/FONT] [1] 0.524508 [/FONT] Slightly lower to-hit, but of course more attacks. With the 16.5 damage per attack, that's [FONT=courier new]dph <- 16.5 daily_attacks <- 3 * N total_damage <- daily_attacks * to_hit * dph total_damage [1] 597.1524 DPR <- total_damage / 20 [1] 29.85762 [/FONT] So, like I said, about 1.5 below what we'd get if we had exactly the "right" number of uses of precision attack every rest. For other ACs, here's what the table looks like with 7 rounds per short rest. [FONT=courier new] total_per_sr DPR 11 305.4 43.6 12 285.6 40.8 13 265.8 38.0 14 246.0 35.1 15 226.2 32.3 16 206.4 29.5 17 186.6 26.7 18 166.8 23.8 19 147.0 21.0 20 127.2 18.2[/FONT] I'm not sure what you're saying here... Precision attack is responsible for more than half of the fighter's edge over the rogue, which we now know how to quantify (thanks, by the way, for spurring me to think about this!). Without superiority dice, the fighter's DPR (vs AC 16) over 20 rounds, assuming three action surges, is just 23 * 3 * 0.40 * 16.5 / 20 = 22.8 which is only about 27% better than the rogue if we assume the rogue gets advantage 75% of the time. (The fighter will get advantage from party members some of the time too, so it's a bigger edge than that, but y'know) [/QUOTE]
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