Reply to thread

Message

<blockquote data-quote="theNater" data-source="post: 4255530" data-attributes="member: 62560">You asked for it, you got it.Assumptions:&nbsp; Using a single fair die, whether a d4, d6, d8, d10, or d12, we will roll it up to four times.&nbsp; We only take a second roll if the first was maximal, we only take a third roll if the second roll was taken and maximal, and we only take a fourth roll if the third roll was taken and maximal.&nbsp; Then we add together the results of all rolls we took and that is our value for the roll.Procedures:&nbsp; We use the expected value formula to find the expected value of our rolls.&nbsp; The expected value is the sum of the products of the possible values with their probabilities.&nbsp; The formula as it applies to the d4 is in the spoiler block for those who are curious.[sblock] (1*(1/4))+(2*(1/4))+(3*(1/4))+(4*0)+(5*(1/16))+(6*(1/16))+(7*(1/16))+(8*0)+(9*(1/64)+(10*(1/64))+(11*(1/64))+(12*0)+(13*(1/256))+(14*(1/256))+(15*(1/256))+(16*(1/256))Note that 4, 8, and 12 do not contribute as they are totals we cannot possibly achieve by rolling a d4 in this manner.[/sblock]Results:&nbsp; The approximate expected values of the various die types are as follows:d4: 3.32d6: 4.20d8: 5.14d10: 6.11d12: 7.09Conclusions:&nbsp; Under this rolling method, each time we step up the die type we increase the expected value by a little less than 1.&nbsp; The larger die types have higher expected values, but not massively higher.</blockquote>

[QUOTE="theNater, post: 4255530, member: 62560"] You asked for it, you got it. Assumptions: Using a single fair die, whether a d4, d6, d8, d10, or d12, we will roll it up to four times. We only take a second roll if the first was maximal, we only take a third roll if the second roll was taken and maximal, and we only take a fourth roll if the third roll was taken and maximal. Then we add together the results of all rolls we took and that is our value for the roll. Procedures: We use the expected value formula to find the expected value of our rolls. The expected value is the sum of the products of the possible values with their probabilities. The formula as it applies to the d4 is in the spoiler block for those who are curious. [sblock] (1*(1/4))+(2*(1/4))+(3*(1/4))+(4*0)+(5*(1/16))+(6*(1/16))+(7*(1/16))+(8*0)+(9*(1/64)+(10*(1/64))+(11*(1/64))+(12*0)+(13*(1/256))+(14*(1/256))+(15*(1/256))+(16*(1/256)) Note that 4, 8, and 12 do not contribute as they are totals we cannot possibly achieve by rolling a d4 in this manner.[/sblock] Results: The approximate expected values of the various die types are as follows: d4: 3.32 d6: 4.20 d8: 5.14 d10: 6.11 d12: 7.09 Conclusions: Under this rolling method, each time we step up the die type we increase the expected value by a little less than 1. The larger die types have higher expected values, but not massively higher. [/QUOTE]

Verification