1. We have added a Gift Upgrades feature that allows you to gift an account upgrade to another member, just in time for the holiday season. You can see the gift option when going to the Account Upgrades screen, or on any user profile screen.
    Dismiss Notice

Let's discuss Mathematics

Discussion in 'Science & Technology' started by ParadigmShifter, Mar 16, 2009.

  1. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
    Meh, applied maths, too useful ;)
     
  2. dutchfire

    dutchfire Deity Retired Moderator

    Joined:
    Jan 5, 2006
    Messages:
    14,106
    Location:
    -
    Well, being applied is the entire point of Calculus.
     
  3. Truronian

    Truronian Quite unfamiliar Retired Moderator

    Joined:
    Aug 3, 2005
    Messages:
    9,869
    Location:
    Near Cornwall
    I like similarity solutions... they're great for turning unsolvable PDEs into unsolvable ODEs.
     
  4. Atticus

    Atticus Deity Retired Moderator

    Joined:
    Aug 20, 2006
    Messages:
    3,666
    Location:
    Helsinki, Finland
    Are there any probability guys here?

    Friend of mine was asked what is the expected value when you cast three dice, disregard the smallest result and add the two other together (if there's two or three dice with the smallest result, you disregard only one).

    For example if you cast 1-4-6, the result is 10.
    If you cast 2-2-4, the result is 6.
    If you cast 3-3-3, the result is 6.

    This is of course very easy to calculate if you go through all the possibilities of the smallest die, and I need no help for that. Instead I want to know if there's easier way to do this, something involving less calculating. Does anybody know?
     
  5. dutchfire

    dutchfire Deity Retired Moderator

    Joined:
    Jan 5, 2006
    Messages:
    14,106
    Location:
    -
    Hmm, I don't see any obvious simplification here. So I'd probably just calculate all the possibilities in excel.
     
  6. Atticus

    Atticus Deity Retired Moderator

    Joined:
    Aug 20, 2006
    Messages:
    3,666
    Location:
    Helsinki, Finland
    No sportsmanship in that solution, and that's why I'm asking. It's probably easier to use brute force, but much much more boring.

    And, GOOD GOD, EXCEL? I can see you're physicist... ;)
     
  7. dutchfire

    dutchfire Deity Retired Moderator

    Joined:
    Jan 5, 2006
    Messages:
    14,106
    Location:
    -
    Sportsmanship? I can see you're not a physicist ;)
    I'll think about it some more, but it'd only take 5 minutes in Excel.
     
  8. Atticus

    Atticus Deity Retired Moderator

    Joined:
    Aug 20, 2006
    Messages:
    3,666
    Location:
    Helsinki, Finland
    Well no need to think it more, I didn't bother either.... :D

    It just feels like there must be some simpler way, and that people who know stochastics better know it.
     
  9. dutchfire

    dutchfire Deity Retired Moderator

    Joined:
    Jan 5, 2006
    Messages:
    14,106
    Location:
    -
    Okay, so I made tables with the first throw fixed.
    Results:
    Spoiler :

    Sum of all options with first throw:
    252
    263
    283
    310
    342
    377

    Difference of these sums:
    11
    20
    27
    32
    35

    Difference of difference
    9
    7
    5
    3

    Difference of difference of difference :)))
    -2
    -2
    -2
    Constant :)
    I believe that could help to make a formula, but I forgot how it works.

    With the first throw is 1, the expected value is obviously 7. If you increase the first throw from 1 to 2, you change 6+6-1=6+5=11 fields, adding 1 to them, giving an expected value of 7+11/36. Then from 2 to 3, you change 6+6+6+6-4=20 fields...
    So you change 12n-n^2 fields when you increase the first throw from n to n+1.
    So the expected values are 7, 7+12-1, 7+12-1+24-2^2 etc. Take the average off these 6 values, and you get 8,46.

    So the expected value if the first throw is n: En= 7 + (sum(i=1 -> n) 12n-n^2)/36
    And then your total expected value is the average of these values, might be a nice formula for that too.
     
  10. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
    I'd enumerate the possibilities as well.

    But here's a (geometric argument) sketch:

    I'll do the 2 dice case first while I work out what's going on...

    P[1 is lowest] = P[A=1 or B=1] = 11/36
    P[2 is lowest] = P[A=2| B>=2 or B=2|A>=2] = 9/36
    P[3 is lowest] = 7/36
    P[4 is lowest] = 5/36
    P[5 is lowest] = 3/36
    P[6 is lowest] = 1/36

    This is just the area of an L-shaped slice in a 6x6 grid, easy to visualise.

    For the 3D case we have a corner shape for the probabilities, so

    P[1 is lowest] = 36 + 30 + 25 / 216 = 91/216
    P[2 is lowest] = 25 + 20 + 16 / 216 = 61/216
    P[3 is lowest] = 16 + 12 + 9 / 216 = 37/216
    P[4 is lowest] = 9 + 6 + 4/216 = 19/216
    P[5 is lowest] = 4 + 2 + 1 = 7/216
    P[6 is lowest] = 1/216

    Then you need to use these weights to multiply the expected roll on the other dice, which is 7 when 1 is the lowest, 8 when 2 is lowest, 9 when 3 is lowest, 10 when 4 is lowest, 11 when 5 is lowest and 12 when 6 is lowest.

    So the answer is:

    91*7/216 + 61*8/216 + 37*9/216 + 19*10/216 + 7*11/216 + 1*12/216

    = (91*7 +61*8 + 37*9 + 19*10 + 7*11 + 1*12)/216
    = (637 + 488 + 333 + 190 + 77 + 12)/216
    = 1737/216 = 8.0416666.....

    I think ;)

    EDIT: Looks like I disagree with dutchfire... I used windows calculator though...
     
  11. dutchfire

    dutchfire Deity Retired Moderator

    Joined:
    Jan 5, 2006
    Messages:
    14,106
    Location:
    -
    Edit: My answer is 1827/216
     
  12. Mise

    Mise isle of lucy

    Joined:
    Apr 13, 2004
    Messages:
    28,623
    Location:
    London, UK
    I get 8.46 too. It's a pretty curve. EDIT: I also used Excel.
     
  13. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
    Can you see a faw in my argument though... maybe I just got some of the sums wrong...
     
  14. dutchfire

    dutchfire Deity Retired Moderator

    Joined:
    Jan 5, 2006
    Messages:
    14,106
    Location:
    -
    I think the weighing you do might be fishy. I thought about that first too, but deemed it too complicated. :)

    The sum of your Ps is 1, so that part is right.
     
  15. dutchfire

    dutchfire Deity Retired Moderator

    Joined:
    Jan 5, 2006
    Messages:
    14,106
    Location:
    -
    For example, when 5 is the lowest, these are the possibilities:
    10,11,11,12,11,12,12,12
    Giving an average of 11,38, not 11.
     
  16. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
    I'll try and recalc the P[x is lowest] working backwards:

    P[6 is lowest] = 1/216. Obvious.
    P[5 is lowest] = P[5 or 6 is lowest] - P[6 is lowest] = 2*2*2/216 - 1/216 = 7/216
    P[4 is lowest] = 3*3*3/216 - 2*2*2/216 = 19/216
    P[3 is lowest] = 4*4*4/216 - 3*3*3/216 = 37/216
    P[2 is lowest] = 5*5*5/216 - 4*4*4/216 = 61/216
    P[1 is lowest] = 216/216 - 5*5*5/216 = 91/216

    So those weights look right to me???

    EDIT: Ah OK it was the expected value on the remaining dice I got wrong.
     
  17. dutchfire

    dutchfire Deity Retired Moderator

    Joined:
    Jan 5, 2006
    Messages:
    14,106
    Location:
    -
    Sorry, I meant these
     
  18. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
    Hold on, when 5 is lowest we get these possibilities:

    {5,5,5} so 10
    {5,5,6} so 11
    {5,6,5} so 11
    {5,6,6} so 12
    {6,5,5} so 11
    {6,5,6} so 12
    {6.6.5} so 12
    {6,6,6} so 12

    so EV is (10 + 3*11 + 4*12) / 8 = 91/8 = 11.375

    Yep, I wasn't taking the conditional probability into account in the final step.
     
  19. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
    At least I had a go without cheating ;)
     
  20. dutchfire

    dutchfire Deity Retired Moderator

    Joined:
    Jan 5, 2006
    Messages:
    14,106
    Location:
    -
    And I got the answer. :p
    Have we found the difference between mathematicians and physicists? :p

    So the final answer is:
    7+sumn=1->6 sumi=1->n (12n-n^2)/216

    ?
     

Share This Page