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Math Help - Drop Rate Inspired.

  1. #1
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    Drop Rate Inspired.

    OK so i was playing a RPG earlier today and was astonished at how many people couldn't understand that drop rates are not linear so I decided to create a simple formula as to how to determine %chance to drop after each kill.

    I started with this:
    (Note)
    a: percent to drop
    ptd: means percent to drop (the acctual words not the number)
    k:kill
    (End Note)


    ptd 1k: a
    ptd 2k: a+(1-a)a= 2a-a^2
    ptd 3k: a+(1-a)a+[1-(a+(1-a)a]a
    =2a-a^2+[1-2a+a^2)]a
    =2a-a^2+a-2a^2+a^3
    =3a-3a^2+a^3
    ptd 4k:4a-6a^2+4a^3-a^4
    ptd 5k:5a-10a^2+10a^3-5a^4+a5

    Now i know there's a pattern in there and I'm sure it has something to do with geometric series but I just cant define the pattern.

    (I'm sure there's an easier way to do this question than my method and I would love to know it so I could give them a formula real fast but I am also curious as to how to find the pattern in these series)

    ^^Sorry for bad grammer up there if you didnt understand plz ask me to explain I will be looking back at this every hour or so.
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  2. #2
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    Quote Originally Posted by Frggr View Post
    OK so i was playing a RPG earlier today and was astonished at how many people couldn't understand that drop rates are not linear so I decided to create a simple formula as to how to determine %chance to drop after each kill.

    I started with this:
    (Note)
    a: percent to drop
    ptd: means percent to drop (the acctual words not the number)
    k:kill
    (End Note)


    ptd 1k: a
    ptd 2k: a+(1-a)a= 2a-a^2
    ptd 3k: a+(1-a)a+[1-(a+(1-a)a]a
    =2a-a^2+[1-2a+a^2)]a
    =2a-a^2+a-2a^2+a^3
    =3a-3a^2+a^3
    ptd 4k:4a-6a^2+4a^3-a^4
    ptd 5k:5a-10a^2+10a^3-5a^4+a5

    Now i know there's a pattern in there and I'm sure it has something to do with geometric series but I just cant define the pattern.

    (I'm sure there's an easier way to do this question than my method and I would love to know it so I could give them a formula real fast but I am also curious as to how to find the pattern in these series)

    ^^Sorry for bad grammer up there if you didnt understand plz ask me to explain I will be looking back at this every hour or so.
    So in a nutshell, you want the rule for generating the following sequence:

    s_1 = a

    s_2 = 2a - a^2

    s_3 = 3a - 3a^2 + a^3

    s_4 = 4a-6a^2+4a^3-a^4

    s_5 = 5a-10a^2+10a^3-5a^4+a^5

    Correct?
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    So in a nutshell, you want the rule for generating the following sequence:

    s_1 = a

    s_2 = 2a - a^2

    s_3 = 3a - 3a^2 + a^3

    s_4 = 4a-6a^2+4a^3-a^4

    s_5 = 5a-10a^2+10a^3-5a^4+a^5

    Correct?

    Correct -- My bad making it more confusing than it should be
    Last edited by Frggr; September 6th 2009 at 09:50 PM.
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  4. #4
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    Quote Originally Posted by mr fantastic View Post
    So in a nutshell, you want the rule for generating the following sequence:

    s_1 = a

    s_2 = 2a - a^2

    s_3 = 3a - 3a^2 + a^3

    s_4 = 4a-6a^2+4a^3-a^4

    s_5 = 5a-10a^2+10a^3-5a^4+a^5

    Correct?
    Just eyeballing it, it appears that

    s_n = 1 - (1 - a)^n = \sum_{i=1}^n (-1)^{n+1} \binom{n}{i} a^i

    where

    \binom{n}{i} = \frac{n!}{i! \; (n-1)!}

    is a binomial coefficient.
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