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On Die Averages and Hit Points in 5e
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<blockquote data-quote="NotAYakk" data-source="post: 8314741" data-attributes="member: 72555"><p>Reroll 1 on first roll makes 1dX become:</p><p>1/X chance of 1dX (average (X+1)/2 = X/2 + 0.5)</p><p>(X-1)/X chance of 1d(X-1)+1; average 1 + (X)/2 = X/2 + 0.5 + 0.5</p><p>So you get X/2+0.5 + 1/2 (X-2)/(X)</p><p></p><p>For 1d6 this is +0.33</p><p>For 1d8 this is +0.38</p><p>For 1d10 this is +0.4</p><p>For 1d12 this is +0.42</p><p></p><p>Reroll 1 always adds +1/2 to the average.</p><p></p><p>Reroll 1 or 2 on first roll makes 1dX become:</p><p>2/X chance of 1dX (average (X+1)/2 = X/2 + 0.5)</p><p>(X-2)/X chance of 1d(X-2)+2. (average 2 + (X-1)/2 = X/2+0.5+1</p><p>So you get X/2+0.5 + (X-2)/X</p><p></p><p>For 1d6 this is +0.67</p><p>For 1d8 this is +0.75</p><p>For 1d10 this is +0.8</p><p>For 1d12 this is +0.83</p><p></p><p>Reroll 1 always and reroll 2 on first roll is 1+1d(X-1) with reroll 1 on first roll.</p><p></p><p>Which is (X-1)/2 + 0.5 + 1/2 (X-3)/(X-1) + 1</p><p>= X/2 + 0.5 + <strong>0.5 + 1/2 (X-3)/(X-1)</strong></p><p></p><p>For a d6 this is +0.8.</p><p>For a d8 this is +0.85</p><p>For a d10 this is +0.89</p><p>For a d12 this is +0.91</p><p></p><p>Reroll 1s and 2s always adds +1 to your average in every case (it is 2+1d(X-2)).</p><p></p><p>---</p><p></p><p>The alternate-system where Con bonus gives you rerolls might work like this.</p><p></p><p>A bonus of +X con means you always reroll HD rolls of X or below. If your bonus exceeds the die size, just use max.</p><p></p><p>A penalty of -1 means you reroll max HD rolls; each additional penalty means you also reroll the next highest roll. Never reroll 1s.</p><p></p><p>In this system, a +1/-1 con is worth 1/2 of a HP/level, about half the impact it has normally. It doesn't work with non-rolling HP however.</p><p></p><p>If you mix it with the "reroll all of your HD" I think it becomes a bit fun. I'd do the "if your con bonus changes, you gain or lose (level * con bonus change)/2 immediately for the corner case, which works unless you have a con of +6 on a d6 HD.</p></blockquote><p></p>
[QUOTE="NotAYakk, post: 8314741, member: 72555"] Reroll 1 on first roll makes 1dX become: 1/X chance of 1dX (average (X+1)/2 = X/2 + 0.5) (X-1)/X chance of 1d(X-1)+1; average 1 + (X)/2 = X/2 + 0.5 + 0.5 So you get X/2+0.5 + 1/2 (X-2)/(X) For 1d6 this is +0.33 For 1d8 this is +0.38 For 1d10 this is +0.4 For 1d12 this is +0.42 Reroll 1 always adds +1/2 to the average. Reroll 1 or 2 on first roll makes 1dX become: 2/X chance of 1dX (average (X+1)/2 = X/2 + 0.5) (X-2)/X chance of 1d(X-2)+2. (average 2 + (X-1)/2 = X/2+0.5+1 So you get X/2+0.5 + (X-2)/X For 1d6 this is +0.67 For 1d8 this is +0.75 For 1d10 this is +0.8 For 1d12 this is +0.83 Reroll 1 always and reroll 2 on first roll is 1+1d(X-1) with reroll 1 on first roll. Which is (X-1)/2 + 0.5 + 1/2 (X-3)/(X-1) + 1 = X/2 + 0.5 + [b]0.5 + 1/2 (X-3)/(X-1)[/b] For a d6 this is +0.8. For a d8 this is +0.85 For a d10 this is +0.89 For a d12 this is +0.91 Reroll 1s and 2s always adds +1 to your average in every case (it is 2+1d(X-2)). --- The alternate-system where Con bonus gives you rerolls might work like this. A bonus of +X con means you always reroll HD rolls of X or below. If your bonus exceeds the die size, just use max. A penalty of -1 means you reroll max HD rolls; each additional penalty means you also reroll the next highest roll. Never reroll 1s. In this system, a +1/-1 con is worth 1/2 of a HP/level, about half the impact it has normally. It doesn't work with non-rolling HP however. If you mix it with the "reroll all of your HD" I think it becomes a bit fun. I'd do the "if your con bonus changes, you gain or lose (level * con bonus change)/2 immediately for the corner case, which works unless you have a con of +6 on a d6 HD. [/QUOTE]
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