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<blockquote data-quote="kerbarian" data-source="post: 2966233" data-attributes="member: 40393">Since there was a decent amount of consensus on that number, I went through and reviewed my code again -- found one more issue.&nbsp; I was only allowing a reroll if the max score was &lt;= 13 and the sum of modifiers was &lt;= 0, instead of if either one is true.&nbsp; After fixing that, I get a number consistent with the others in that thread -- 30.4769.&nbsp; Note that this is an analytical/deterministic result, not stochastic.With that final correction, here's a review of the average point buy costs for various dice rolling methods:Average Characters (3d6, reroll if max &lt; 12 or modifiers &lt; -2):&nbsp; 19.5109 (26.3% chance of reroll)Standard rolling (4d6, drop lowest, reroll if max &lt; 14 or modifiers &lt; +1):&nbsp; 30.4769 (12.9% chance of reroll)Standard + reroll ones (4d6, drop lowest, reroll 1s, reroll if max &lt; 14 or modifiers &lt; +1):&nbsp; 37.7441 (2.4% chance of reroll)High-Powered Characters (5d6, drop lowest 2, reroll if max &lt; 15 or modifiers &lt; +2):&nbsp; 39.1323 (6.6% chance of reroll)Also, I cleaned up and consolidated the code, and I added comments.&nbsp; So it's available for double-checking, or if anyone wants to use it as a basis for other calculations.&nbsp; The program is in C++, and it runs in about 1.3 seconds on a 2GHz machine whe compiled using gcc -O2.&nbsp; There's not much code involved, but it's bulked up to about double its raw size due to comments.This is for the straight 4d6-drop-lowest version, but it's easy to change with the #defines at the top (rerolling 1s is done by using 5-sided dice with a die bonus of 1):[CODE]#include &lt;math.h&gt;#include &lt;memory.h&gt;#include &lt;stdio.h&gt;#include &lt;algorithm&gt;#define NUM_DICE 4#define NUM_SIDES 6#define DIE_BONUS 0#define REROLL_MAX_SCORE_THRESHOLD 14#define REROLL_MODIFIER_SUM_THRESHOLD 1// calculate the probability of each score and fill it into _score_probvoid calculate_score_probs(double* _score_prob){&nbsp; &nbsp; memset(_score_prob, 0, sizeof(double)*19);&nbsp; &nbsp; // iterate through all possible sets of dice rolls&nbsp; &nbsp; int num_possibilities = (int)pow(NUM_SIDES, NUM_DICE);&nbsp; &nbsp; for(int i=1; i&lt;num_possibilities; i++) {&nbsp; &nbsp; &nbsp; &nbsp; int die[NUM_DICE];&nbsp; &nbsp; &nbsp; &nbsp; memset(die, 0, sizeof(int)*NUM_DICE);&nbsp; &nbsp; &nbsp; &nbsp; // turn this index into individual die roll numbers&nbsp; &nbsp; &nbsp; &nbsp; int i2 = i;&nbsp; &nbsp; &nbsp; &nbsp; for(int j=0; j&lt;NUM_DICE; j++) {&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; die[j] = i2 % NUM_SIDES + 1 + DIE_BONUS;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; i2 /= NUM_SIDES;&nbsp; &nbsp; &nbsp; &nbsp; }&nbsp; &nbsp; &nbsp; &nbsp; // sort the dice so that the highest rolls are first in the array&nbsp; &nbsp; &nbsp; &nbsp; std::sort(die, die+NUM_DICE);&nbsp; &nbsp; &nbsp; &nbsp; std::reverse(die, die+NUM_DICE);&nbsp; &nbsp; &nbsp; &nbsp; // increment the probability for the score that's the sum of the highest 3 dice&nbsp; &nbsp; &nbsp; &nbsp; _score_prob[die[0]+die[1]+die[2]] += 1.0/num_possibilities;&nbsp; &nbsp; }}int main(){&nbsp; &nbsp; // score_prob is the probability of rolling each score. The score is the index into&nbsp; &nbsp; // the array, so the first 3 entries will be empty.&nbsp; &nbsp; double score_prob[19];&nbsp; &nbsp; calculate_score_probs(score_prob);&nbsp; &nbsp; // score_cost is an array of the point costs for each score, indexed the same way&nbsp; &nbsp; int score_cost[19] = {0, 0, 0, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 8, 10, 13, 16};&nbsp; &nbsp; // prob_of_cost is the probability of a set of scores having a specific total point&nbsp; &nbsp; // cost (ignoring score sets that can be rerolled).&nbsp; The point cost + 24 is the index&nbsp; &nbsp; // into the array (total possible range of -24 to 96).&nbsp; &nbsp; double prob_of_cost[121];&nbsp; &nbsp; memset(prob_of_cost, 0, sizeof(double)*121);&nbsp; &nbsp; // total prob is the probability of rolling a set of scores that can't be rerolled.&nbsp; &nbsp; double total_prob = 0;&nbsp; &nbsp; // iterate through all possible sets of scores.&nbsp; There are 16 possible scores (3-18),&nbsp; &nbsp; // so the total number of sets of scores is 16^6 = 16777216.&nbsp; &nbsp; for(int i=0; i&lt;16777216; i++)&nbsp; &nbsp; {&nbsp; &nbsp; &nbsp; &nbsp; bool max_score_over_threshold = false;&nbsp; &nbsp; &nbsp; &nbsp; int modifier_sum = 0;&nbsp; &nbsp; &nbsp; &nbsp; double prob = 1;&nbsp; &nbsp; &nbsp; &nbsp; int cost = 0;&nbsp; &nbsp; &nbsp; &nbsp; // turn the index into a set of individual scores&nbsp; &nbsp; &nbsp; &nbsp; int i2 = i;&nbsp; &nbsp; &nbsp; &nbsp; for(int j=0; j &lt; 6; j++) {&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; int score = i2 % 16 + 3;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; i2 /= 16;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; // for each score, record if it's over 13 or and its effect on the sum of&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; // the modifiers, so we can decide if this score set can be rerolled.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; if(score &gt;= REROLL_MAX_SCORE_THRESHOLD)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; max_score_over_threshold = true;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; modifier_sum += score/2 - 5;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; // after each roll, multiply the probability of this score set by the&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; // probability of rolling the most recent score.&nbsp; By the end of the set,&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; // we'll have the overall probability of the entire score set.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; prob *= score_prob[score];&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; // likewise, add the point cost of the current score into the total for&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; // this set.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; cost += score_cost[score];&nbsp; &nbsp; &nbsp; &nbsp; }&nbsp; &nbsp; &nbsp; &nbsp; // only count this score set if it can't be rerolled&nbsp; &nbsp; &nbsp; &nbsp; if(max_score_over_threshold &amp;&amp; modifier_sum &gt;= REROLL_MODIFIER_SUM_THRESHOLD) {&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; prob_of_cost[cost+24] += prob;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; total_prob += prob;&nbsp; &nbsp; &nbsp; &nbsp; }&nbsp; &nbsp; }&nbsp; &nbsp; // add up the cost of each score set, weighted by the probability of that score&nbsp; &nbsp; // set occurring.&nbsp; &nbsp; double total_cost = 0;&nbsp; &nbsp; for(int i=0; i&lt;=121; i++)&nbsp; &nbsp; &nbsp; &nbsp; total_cost += (i-24) * prob_of_cost[i];&nbsp; &nbsp; printf(&quot;score sets that can be rerolled:&nbsp; %.4f%\n&quot;, (1 - total_prob)*100);&nbsp; &nbsp; // the overall EV is the (weighted) average cost of non-rerollable score sets&nbsp; &nbsp; // divided by the chance of rolling a non-rerollable score set in the first place.&nbsp; &nbsp; printf(&quot;average point cost after rerolls:&nbsp; %.4f\n&quot;, total_cost/total_prob);&nbsp; &nbsp; return 0;}[/CODE]</blockquote>

[QUOTE="kerbarian, post: 2966233, member: 40393"] Since there was a decent amount of consensus on that number, I went through and reviewed my code again -- found one more issue. I was only allowing a reroll if the max score was <= 13 [i]and[/i] the sum of modifiers was <= 0, instead of if either one is true. After fixing that, I get a number consistent with the others in that thread -- 30.4769. Note that this is an analytical/deterministic result, not stochastic. With that final correction, here's a review of the average point buy costs for various dice rolling methods: Average Characters (3d6, reroll if max < 12 or modifiers < -2): 19.5109 (26.3% chance of reroll) Standard rolling (4d6, drop lowest, reroll if max < 14 or modifiers < +1): 30.4769 (12.9% chance of reroll) Standard + reroll ones (4d6, drop lowest, reroll 1s, reroll if max < 14 or modifiers < +1): 37.7441 (2.4% chance of reroll) High-Powered Characters (5d6, drop lowest 2, reroll if max < 15 or modifiers < +2): 39.1323 (6.6% chance of reroll) Also, I cleaned up and consolidated the code, and I added comments. So it's available for double-checking, or if anyone wants to use it as a basis for other calculations. The program is in C++, and it runs in about 1.3 seconds on a 2GHz machine whe compiled using gcc -O2. There's not much code involved, but it's bulked up to about double its raw size due to comments. This is for the straight 4d6-drop-lowest version, but it's easy to change with the #defines at the top (rerolling 1s is done by using 5-sided dice with a die bonus of 1): [CODE]#include <math.h> #include <memory.h> #include <stdio.h> #include <algorithm> #define NUM_DICE 4 #define NUM_SIDES 6 #define DIE_BONUS 0 #define REROLL_MAX_SCORE_THRESHOLD 14 #define REROLL_MODIFIER_SUM_THRESHOLD 1 // calculate the probability of each score and fill it into _score_prob void calculate_score_probs(double* _score_prob) { memset(_score_prob, 0, sizeof(double)*19); // iterate through all possible sets of dice rolls int num_possibilities = (int)pow(NUM_SIDES, NUM_DICE); for(int i=1; i<num_possibilities; i++) { int die[NUM_DICE]; memset(die, 0, sizeof(int)*NUM_DICE); // turn this index into individual die roll numbers int i2 = i; for(int j=0; j<NUM_DICE; j++) { die[j] = i2 % NUM_SIDES + 1 + DIE_BONUS; i2 /= NUM_SIDES; } // sort the dice so that the highest rolls are first in the array std::sort(die, die+NUM_DICE); std::reverse(die, die+NUM_DICE); // increment the probability for the score that's the sum of the highest 3 dice _score_prob[die[0]+die[1]+die[2]] += 1.0/num_possibilities; } } int main() { // score_prob is the probability of rolling each score. The score is the index into // the array, so the first 3 entries will be empty. double score_prob[19]; calculate_score_probs(score_prob); // score_cost is an array of the point costs for each score, indexed the same way int score_cost[19] = {0, 0, 0, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 8, 10, 13, 16}; // prob_of_cost is the probability of a set of scores having a specific total point // cost (ignoring score sets that can be rerolled). The point cost + 24 is the index // into the array (total possible range of -24 to 96). double prob_of_cost[121]; memset(prob_of_cost, 0, sizeof(double)*121); // total prob is the probability of rolling a set of scores that can't be rerolled. double total_prob = 0; // iterate through all possible sets of scores. There are 16 possible scores (3-18), // so the total number of sets of scores is 16^6 = 16777216. for(int i=0; i<16777216; i++) { bool max_score_over_threshold = false; int modifier_sum = 0; double prob = 1; int cost = 0; // turn the index into a set of individual scores int i2 = i; for(int j=0; j < 6; j++) { int score = i2 % 16 + 3; i2 /= 16; // for each score, record if it's over 13 or and its effect on the sum of // the modifiers, so we can decide if this score set can be rerolled. if(score >= REROLL_MAX_SCORE_THRESHOLD) max_score_over_threshold = true; modifier_sum += score/2 - 5; // after each roll, multiply the probability of this score set by the // probability of rolling the most recent score. By the end of the set, // we'll have the overall probability of the entire score set. prob *= score_prob[score]; // likewise, add the point cost of the current score into the total for // this set. cost += score_cost[score]; } // only count this score set if it can't be rerolled if(max_score_over_threshold && modifier_sum >= REROLL_MODIFIER_SUM_THRESHOLD) { prob_of_cost[cost+24] += prob; total_prob += prob; } } // add up the cost of each score set, weighted by the probability of that score // set occurring. double total_cost = 0; for(int i=0; i<=121; i++) total_cost += (i-24) * prob_of_cost[i]; printf("score sets that can be rerolled: %.4f%\n", (1 - total_prob)*100); // the overall EV is the (weighted) average cost of non-rerollable score sets // divided by the chance of rolling a non-rerollable score set in the first place. printf("average point cost after rerolls: %.4f\n", total_cost/total_prob); return 0; }[/CODE] [/QUOTE]

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