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Point Buy vs Rolling for Stats
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<blockquote data-quote="Harzel" data-source="post: 7260381" data-attributes="member: 6857506"><p>Update to this post.</p><p></p><p></p><p></p><p>Also, @<em><strong><u><a href="http://www.enworld.org/forum/member.php?u=6801845" target="_blank">Oofta</a></u></strong></em>’s numbers again for reference.</p><p></p><p></p><p></p><p>Below are the frequency distributions for each ranked value in attribute arrays generated by 6 x 4d6k3 – that is, highest value, 2[SUP]nd[/SUP] highest, …, down to lowest. Below each distribution are the mean, median, and mode, which are all identical, excepting one case. The number of Big Surprises is, well, pretty much 0. Note that Oofta’s sample means are good to ‘only’ five digits; yeah, that there sampling worked pretty good.</p><p> </p><p>The attached spreadsheet also has tables of cumulative probabilities both ascending and descending. (For convenience. They are equivalent to the values in the table below.) A few data points from those:</p><ul> <li data-xf-list-type="ul">Probability of # of values ≤ 5: 1 – 7%</li> <li data-xf-list-type="ul">Probability of # of values ≤ 7: 1 – 30%; 2 – 4%</li> <li data-xf-list-type="ul">Probability of # of values ≤ 9: 1 – 69%; 2 – 28%; 3 – 7%</li> <li data-xf-list-type="ul">Probability of # of values ≥ 14: 1 – 93%; 2 – 69%; 3 – 36%; 4 – 12%</li> <li data-xf-list-type="ul">Probability of # of values ≥ 15: 1 – 72%; 2 – 42%; 3 – 14%; 4 – 3%</li> <li data-xf-list-type="ul">Probability of # of values ≥ 16: 1 – 57%; 2 – 18%; 3 – 3%</li> <li data-xf-list-type="ul">Probability of # of values ≥ 17: 1 – 30%; 2 – 4%</li> <li data-xf-list-type="ul">Probability of # of values = 18: 1 – 9%</li> </ul><p>FWIW, my sort of rough feeling based on all of this is that there are maybe 10-12% of the arrays that I would be somewhat loathe to play as a player and for which as a DM I would feel obliged to let the player reroll. On the other end, I think there are maybe around 20% which would feel like an embarrassment of riches as a player and would look out of whack as a DM. Overall, I think this makes me more favorably inclined than I was toward point buy even though both I and my players like rolling for stats just as a mini-game.</p><p></p><p>[TABLE="class: grid, width: 600"]</p><p>[TR]</p><p>[TD="colspan: 3"]<strong>Probabilities of value by rank</strong>[/TD]</p><p>[TD][/TD]</p><p>[TD][/TD]</p><p>[TD][/TD]</p><p>[TD][/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>Value</strong>[/TD]</p><p>[TD]<strong>Highest</strong>[/TD]</p><p>[TD]<strong>2nd</strong>[/TD]</p><p>[TD]<strong>3rd</strong>[/TD]</p><p>[TD]<strong>4th</strong>[/TD]</p><p>[TD]<strong>5th</strong>[/TD]</p><p>[TD]<strong>Lowest</strong>[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>3</strong>[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0046[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>4</strong>[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0002[/TD]</p><p>[TD]0.0183[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>5</strong>[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0017[/TD]</p><p>[TD]0.0445[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>6</strong>[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0004[/TD]</p><p>[TD]0.0088[/TD]</p><p>[TD]0.0880[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>7</strong>[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0001[/TD]</p><p>[TD]0.0029[/TD]</p><p>[TD]0.0312[/TD]</p><p>[TD]0.1417[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>8</strong>[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0001[/TD]</p><p>[TD]0.0014[/TD]</p><p>[TD]0.0148[/TD]</p><p>[TD]0.0822[/TD]</p><p>[TD]0.1886[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>9</strong>[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0008[/TD]</p><p>[TD]0.0089[/TD]</p><p>[TD]0.0527[/TD]</p><p>[TD]0.1597[/TD]</p><p>[TD]0.1992[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>10</strong>[/TD]</p><p>[TD]0.0004[/TD]</p><p>[TD]0.0057[/TD]</p><p>[TD]0.0383[/TD]</p><p>[TD]0.1303[/TD]</p><p>[TD]0.2274[/TD]</p><p>[TD]0.1627[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>11</strong>[/TD]</p><p>[TD]0.0028[/TD]</p><p>[TD]0.0273[/TD]</p><p>[TD]0.1085[/TD]</p><p>[TD]0.2204[/TD]</p><p>[TD]0.2290[/TD]</p><p>[TD]0.0973[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>12</strong>[/TD]</p><p>[TD]0.0149[/TD]</p><p>[TD]0.0875[/TD]</p><p>[TD]0.2100[/TD]</p><p>[TD]0.2576[/TD]</p><p>[TD]0.1616[/TD]</p><p>[TD]0.0415[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>13</strong>[/TD]</p><p>[TD]0.0540[/TD]</p><p>[TD]0.1885[/TD]</p><p>[TD]0.2699[/TD]</p><p>[TD]0.1980[/TD]</p><p>[TD]0.0744[/TD]</p><p>[TD]0.0114[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>14</strong>[/TD]</p><p>[TD]0.1340[/TD]</p><p>[TD]0.2685[/TD]</p><p>[TD]0.2217[/TD]</p><p>[TD]0.0942[/TD]</p><p>[TD]0.0206[/TD]</p><p>[TD]0.0018[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>15</strong>[/TD]</p><p>[TD]0.2264[/TD]</p><p>[TD]0.2431[/TD]</p><p>[TD]0.1086[/TD]</p><p>[TD]0.0252[/TD]</p><p>[TD]0.0030[/TD]</p><p>[TD]0.0001[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>16</strong>[/TD]</p><p>[TD]0.2669[/TD]</p><p>[TD]0.1355[/TD]</p><p>[TD]0.0293[/TD]</p><p>[TD]0.0033[/TD]</p><p>[TD]0.0002[/TD]</p><p>[TD]0.0000[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>17</strong>[/TD]</p><p>[TD]0.2073[/TD]</p><p>[TD]0.0392[/TD]</p><p>[TD]0.0033[/TD]</p><p>[TD]0.0002[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>18</strong>[/TD]</p><p>[TD]0.0934[/TD]</p><p>[TD]0.0038[/TD]</p><p>[TD]0.0001[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[TD]0.0000[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>Mode</strong>[/TD]</p><p>[TD]16[/TD]</p><p>[TD]14[/TD]</p><p>[TD]13[/TD]</p><p>[TD]12[/TD]</p><p>[TD]11[/TD]</p><p>[TD]9[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>Mean</strong>[/TD]</p><p>[TD]15.6618863[/TD]</p><p>[TD]14.1742279[/TD]</p><p>[TD]12.9551079[/TD]</p><p>[TD]11.7609295[/TD]</p><p>[TD]10.411357[/TD]</p><p>[TD]8.50408450[/TD]</p><p>[/TR]</p><p>[TR]</p><p>[TD]<strong>Median</strong>[/TD]</p><p>[TD]16[/TD]</p><p>[TD]14[/TD]</p><p>[TD]13[/TD]</p><p>[TD]12[/TD]</p><p>[TD]10[/TD]</p><p>[TD]9[/TD]</p><p>[/TR]</p><p>[/TABLE]</p><p> </p><p>Note: Entries that appear to be 0 are just values < 10^-4. I just thought it was easier to read with fewer digits and I have an irrational aversion to scientific notation.</p></blockquote><p></p>
[QUOTE="Harzel, post: 7260381, member: 6857506"] Update to this post. Also, @[I][B][U][URL="http://www.enworld.org/forum/member.php?u=6801845"]Oofta[/URL][/U][/B][/I]’s numbers again for reference. Below are the frequency distributions for each ranked value in attribute arrays generated by 6 x 4d6k3 – that is, highest value, 2[SUP]nd[/SUP] highest, …, down to lowest. Below each distribution are the mean, median, and mode, which are all identical, excepting one case. The number of Big Surprises is, well, pretty much 0. Note that Oofta’s sample means are good to ‘only’ five digits; yeah, that there sampling worked pretty good. The attached spreadsheet also has tables of cumulative probabilities both ascending and descending. (For convenience. They are equivalent to the values in the table below.) A few data points from those: [LIST] [*]Probability of # of values ≤ 5: 1 – 7% [*]Probability of # of values ≤ 7: 1 – 30%; 2 – 4% [*]Probability of # of values ≤ 9: 1 – 69%; 2 – 28%; 3 – 7% [*]Probability of # of values ≥ 14: 1 – 93%; 2 – 69%; 3 – 36%; 4 – 12% [*]Probability of # of values ≥ 15: 1 – 72%; 2 – 42%; 3 – 14%; 4 – 3% [*]Probability of # of values ≥ 16: 1 – 57%; 2 – 18%; 3 – 3% [*]Probability of # of values ≥ 17: 1 – 30%; 2 – 4% [*]Probability of # of values = 18: 1 – 9% [/LIST] FWIW, my sort of rough feeling based on all of this is that there are maybe 10-12% of the arrays that I would be somewhat loathe to play as a player and for which as a DM I would feel obliged to let the player reroll. On the other end, I think there are maybe around 20% which would feel like an embarrassment of riches as a player and would look out of whack as a DM. Overall, I think this makes me more favorably inclined than I was toward point buy even though both I and my players like rolling for stats just as a mini-game. [TABLE="class: grid, width: 600"] [TR] [TD="colspan: 3"][B]Probabilities of value by rank[/B][/TD] [TD][/TD] [TD][/TD] [TD][/TD] [TD][/TD] [/TR] [TR] [TD][B]Value[/B][/TD] [TD][B]Highest[/B][/TD] [TD][B]2nd[/B][/TD] [TD][B]3rd[/B][/TD] [TD][B]4th[/B][/TD] [TD][B]5th[/B][/TD] [TD][B]Lowest[/B][/TD] [/TR] [TR] [TD][B]3[/B][/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0046[/TD] [/TR] [TR] [TD][B]4[/B][/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0002[/TD] [TD]0.0183[/TD] [/TR] [TR] [TD][B]5[/B][/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0017[/TD] [TD]0.0445[/TD] [/TR] [TR] [TD][B]6[/B][/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0004[/TD] [TD]0.0088[/TD] [TD]0.0880[/TD] [/TR] [TR] [TD][B]7[/B][/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0001[/TD] [TD]0.0029[/TD] [TD]0.0312[/TD] [TD]0.1417[/TD] [/TR] [TR] [TD][B]8[/B][/TD] [TD]0.0000[/TD] [TD]0.0001[/TD] [TD]0.0014[/TD] [TD]0.0148[/TD] [TD]0.0822[/TD] [TD]0.1886[/TD] [/TR] [TR] [TD][B]9[/B][/TD] [TD]0.0000[/TD] [TD]0.0008[/TD] [TD]0.0089[/TD] [TD]0.0527[/TD] [TD]0.1597[/TD] [TD]0.1992[/TD] [/TR] [TR] [TD][B]10[/B][/TD] [TD]0.0004[/TD] [TD]0.0057[/TD] [TD]0.0383[/TD] [TD]0.1303[/TD] [TD]0.2274[/TD] [TD]0.1627[/TD] [/TR] [TR] [TD][B]11[/B][/TD] [TD]0.0028[/TD] [TD]0.0273[/TD] [TD]0.1085[/TD] [TD]0.2204[/TD] [TD]0.2290[/TD] [TD]0.0973[/TD] [/TR] [TR] [TD][B]12[/B][/TD] [TD]0.0149[/TD] [TD]0.0875[/TD] [TD]0.2100[/TD] [TD]0.2576[/TD] [TD]0.1616[/TD] [TD]0.0415[/TD] [/TR] [TR] [TD][B]13[/B][/TD] [TD]0.0540[/TD] [TD]0.1885[/TD] [TD]0.2699[/TD] [TD]0.1980[/TD] [TD]0.0744[/TD] [TD]0.0114[/TD] [/TR] [TR] [TD][B]14[/B][/TD] [TD]0.1340[/TD] [TD]0.2685[/TD] [TD]0.2217[/TD] [TD]0.0942[/TD] [TD]0.0206[/TD] [TD]0.0018[/TD] [/TR] [TR] [TD][B]15[/B][/TD] [TD]0.2264[/TD] [TD]0.2431[/TD] [TD]0.1086[/TD] [TD]0.0252[/TD] [TD]0.0030[/TD] [TD]0.0001[/TD] [/TR] [TR] [TD][B]16[/B][/TD] [TD]0.2669[/TD] [TD]0.1355[/TD] [TD]0.0293[/TD] [TD]0.0033[/TD] [TD]0.0002[/TD] [TD]0.0000[/TD] [/TR] [TR] [TD][B]17[/B][/TD] [TD]0.2073[/TD] [TD]0.0392[/TD] [TD]0.0033[/TD] [TD]0.0002[/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [/TR] [TR] [TD][B]18[/B][/TD] [TD]0.0934[/TD] [TD]0.0038[/TD] [TD]0.0001[/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [TD]0.0000[/TD] [/TR] [TR] [TD][B]Mode[/B][/TD] [TD]16[/TD] [TD]14[/TD] [TD]13[/TD] [TD]12[/TD] [TD]11[/TD] [TD]9[/TD] [/TR] [TR] [TD][B]Mean[/B][/TD] [TD]15.6618863[/TD] [TD]14.1742279[/TD] [TD]12.9551079[/TD] [TD]11.7609295[/TD] [TD]10.411357[/TD] [TD]8.50408450[/TD] [/TR] [TR] [TD][B]Median[/B][/TD] [TD]16[/TD] [TD]14[/TD] [TD]13[/TD] [TD]12[/TD] [TD]10[/TD] [TD]9[/TD] [/TR] [/TABLE] Note: Entries that appear to be 0 are just values < 10^-4. I just thought it was easier to read with fewer digits and I have an irrational aversion to scientific notation. [/QUOTE]
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