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Need someone to check my dice probability results
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<blockquote data-quote="physicscarp" data-source="post: 1867492" data-attributes="member: 7207"><p>Here is the situation:</p><p></p><p>In the game we are playing, characters may have a skill rank ranging from 1 to 20 <img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" class="smilie smilie--sprite smilie--sprite23" alt="(n)" title="Thumbs down (n)" loading="lazy" data-shortname="(n)" />, and to suceed at a task, the player must roll at or below his skill rank on a d20. Therefore the chance of success (S) for an event is S = n/20.</p><p></p><p>Now, during the game, a player may spend a hero point to either...</p><p>1) add +4 to his original skill level or</p><p>2) reroll a failed roll </p><p></p><p>I'm wondering at what skill rank does it become more beneficial to reroll rather than bump the skill rank. Here's what I have worked out, but I'm hoping someone can confirm this for me.</p><p></p><p>For case 1 (bumping the rank by 4), the probability of succeeding becomes S = (n+4)/20. </p><p></p><p>Case 2 is where I am having trouble. Do I treat these events as independent? It doesn't seem like I should. I arranged the results in a sample space and come up with 400 possible results. To find the number of results that would give a success, I found this relationship, <strong>n(40-n)</strong>, so therefore the chance of success would be S = <strong>(n(40-n))/400</strong>. Using this I get the following results...</p><p></p><p>(Rank/% Chance to Succeed Using Reroll)</p><p>1/9.75</p><p>2/19</p><p>3/27.75</p><p>4/36...</p><p></p><p>Comparing this to the results from case 1, it seems that once you have acheived rank 6 or higher it is more beneficial to reroll. </p><p></p><p>Does this make sense to any stats monkeys among us? Thanks for any help you can provide.</p></blockquote><p></p>
[QUOTE="physicscarp, post: 1867492, member: 7207"] Here is the situation: In the game we are playing, characters may have a skill rank ranging from 1 to 20 (n), and to suceed at a task, the player must roll at or below his skill rank on a d20. Therefore the chance of success (S) for an event is S = n/20. Now, during the game, a player may spend a hero point to either... 1) add +4 to his original skill level or 2) reroll a failed roll I'm wondering at what skill rank does it become more beneficial to reroll rather than bump the skill rank. Here's what I have worked out, but I'm hoping someone can confirm this for me. For case 1 (bumping the rank by 4), the probability of succeeding becomes S = (n+4)/20. Case 2 is where I am having trouble. Do I treat these events as independent? It doesn't seem like I should. I arranged the results in a sample space and come up with 400 possible results. To find the number of results that would give a success, I found this relationship, [B]n(40-n)[/B], so therefore the chance of success would be S = [B](n(40-n))/400[/B]. Using this I get the following results... (Rank/% Chance to Succeed Using Reroll) 1/9.75 2/19 3/27.75 4/36... Comparing this to the results from case 1, it seems that once you have acheived rank 6 or higher it is more beneficial to reroll. Does this make sense to any stats monkeys among us? Thanks for any help you can provide. [/QUOTE]
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