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House rule: Extra Attacks for martials at 5,11,17
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<blockquote data-quote="Kinematics" data-source="post: 9143153" data-attributes="member: 6932123"><p>This is the (simplified) formula you used:</p><p></p><p>Base accuracy: 65%</p><p>Miss rate: 35%</p><p></p><p>Chance of landing 0 hits out of 4 attacks = 0.01500625 (rounded to 0.0150, or 1.50%) (not used above)</p><p>Chance of landing 1 hit out of 4 attacks = 0.111475 (rounded to 0.1115, or 11.15%)</p><p>Chance of landing 2 hits out of 4 attacks = 0.3105375 (rounded to 0.3105, or 31.05%)</p><p>Chance of landing 3 hits out of 4 attacks = 0.384475 (rounded to 0.3845, or 38.45%)</p><p>Chance of landing 4 hits out of 4 attacks = 0.17850625 (rounded to 0.1785, or 17.85%)</p><p></p><p>Then look at some of the components. For example:</p><p></p><p>0.1115×0.05×5.5 </p><p></p><p>This is the 11.15% chance of landing 1 hit out of 4, the 5% chance of landing a crit, and the 5.5 damage the crit did. The problem here is that the 5% * 11.15% calculation is not correct. You've already landed your one attack, but multiplying by 5% indicates that only one out of twenty of those hits was a crit. This is incorrect.</p><p></p><p>Put in die roll terms, a 65% hit rate means that you rolled an 8 or better. The 35% miss rate is rolling a 1 through 7. That means there were 13 possible values for you to have rolled, of which one of them is a crit. </p><p></p><p>Thus, given that you know you already hit, the chance of it being a crit is 1 in 13, not 1 in 20.</p><p></p><p>Of course this would mean that you're undervaluing how much crit damage was done, rather than overvaluing as seen in the actual result, so let's continue to the next.</p><p></p><p>0.3105×(0.05^2×11+0.975×5.5)</p><p></p><p>This one is landing two hits, and then tries to combine the value of both hits being crits, and the chance of one hit being a crit. There's still the 1 in 20 vs 1 in 13 error, but in addition to that you've assumed that if you didn't land two crits (0.05^2), that you are almost guaranteed to have landed one (0.975).</p><p></p><p>I'm not sure where you got the 0.975 from. It looks like perhaps a typo versus the 0.9975 that would be all combinations that were not two crits. Even if we keep the assumption of 5% for the crit, this value should be 0.095. This creates a 1.5 point overvaluation before correcting to 1/13.</p><p></p><p>0.3845×(0.05^3×16.5+0.0071×11+0.1354×5.5)</p><p></p><p>The values here are correct for a 1 in 20, but of course need to be corrected to 1 in 13.</p><p></p><p>0.1785×(0.05^4×22+0.00048125×16.5+0.0135×11+0.1715×5.5)</p><p></p><p>0.00048125 appears to be incorrect. It should be 0.000475.</p><p></p><p>So overall you need to subtract 1.5 from the 2.2, leaving you 0.7, but then everything needs to scale up because you should be using 7.69% (1 in 13) instead of 5% (1 in 20).</p><p></p><p>You'd end up with:</p><p></p><p>[plain]0.1115 * (0.0769 * 5.5)</p><p>+ 0.3105 * (0.0769^2 * 11 + 0.1420 * 5.5)</p><p>+ 0.3845 * (0.0769^3 * 16.5 + 0.0164×11 + 0.1966×5.5)</p><p>+ 0.1785 * (0.0769^4 * 22 + 0.0004198×16.5 + 0.0302×11 + 0.242×5.5)</p><p>= 0.1115 * 0.423</p><p>+ 0.3105 * 0.846</p><p>+ 0.3845 * (0.0075 + 0.1801 + 1.0812)</p><p>+ 0.1785 * (0.000769 + 0.00693 + 0.3326 + 1.331)</p><p>= 1.096[/plain]</p><p></p><p>Which, allowing for rounding (I only used the approximate of 1/13), matches the expected 1.1.</p></blockquote><p></p>
[QUOTE="Kinematics, post: 9143153, member: 6932123"] This is the (simplified) formula you used: Base accuracy: 65% Miss rate: 35% Chance of landing 0 hits out of 4 attacks = 0.01500625 (rounded to 0.0150, or 1.50%) (not used above) Chance of landing 1 hit out of 4 attacks = 0.111475 (rounded to 0.1115, or 11.15%) Chance of landing 2 hits out of 4 attacks = 0.3105375 (rounded to 0.3105, or 31.05%) Chance of landing 3 hits out of 4 attacks = 0.384475 (rounded to 0.3845, or 38.45%) Chance of landing 4 hits out of 4 attacks = 0.17850625 (rounded to 0.1785, or 17.85%) Then look at some of the components. For example: 0.1115×0.05×5.5 This is the 11.15% chance of landing 1 hit out of 4, the 5% chance of landing a crit, and the 5.5 damage the crit did. The problem here is that the 5% * 11.15% calculation is not correct. You've already landed your one attack, but multiplying by 5% indicates that only one out of twenty of those hits was a crit. This is incorrect. Put in die roll terms, a 65% hit rate means that you rolled an 8 or better. The 35% miss rate is rolling a 1 through 7. That means there were 13 possible values for you to have rolled, of which one of them is a crit. Thus, given that you know you already hit, the chance of it being a crit is 1 in 13, not 1 in 20. Of course this would mean that you're undervaluing how much crit damage was done, rather than overvaluing as seen in the actual result, so let's continue to the next. 0.3105×(0.05^2×11+0.975×5.5) This one is landing two hits, and then tries to combine the value of both hits being crits, and the chance of one hit being a crit. There's still the 1 in 20 vs 1 in 13 error, but in addition to that you've assumed that if you didn't land two crits (0.05^2), that you are almost guaranteed to have landed one (0.975). I'm not sure where you got the 0.975 from. It looks like perhaps a typo versus the 0.9975 that would be all combinations that were not two crits. Even if we keep the assumption of 5% for the crit, this value should be 0.095. This creates a 1.5 point overvaluation before correcting to 1/13. 0.3845×(0.05^3×16.5+0.0071×11+0.1354×5.5) The values here are correct for a 1 in 20, but of course need to be corrected to 1 in 13. 0.1785×(0.05^4×22+0.00048125×16.5+0.0135×11+0.1715×5.5) 0.00048125 appears to be incorrect. It should be 0.000475. So overall you need to subtract 1.5 from the 2.2, leaving you 0.7, but then everything needs to scale up because you should be using 7.69% (1 in 13) instead of 5% (1 in 20). You'd end up with: [plain]0.1115 * (0.0769 * 5.5) + 0.3105 * (0.0769^2 * 11 + 0.1420 * 5.5) + 0.3845 * (0.0769^3 * 16.5 + 0.0164×11 + 0.1966×5.5) + 0.1785 * (0.0769^4 * 22 + 0.0004198×16.5 + 0.0302×11 + 0.242×5.5) = 0.1115 * 0.423 + 0.3105 * 0.846 + 0.3845 * (0.0075 + 0.1801 + 1.0812) + 0.1785 * (0.000769 + 0.00693 + 0.3326 + 1.331) = 1.096[/plain] Which, allowing for rounding (I only used the approximate of 1/13), matches the expected 1.1. [/QUOTE]
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House rule: Extra Attacks for martials at 5,11,17
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