Please turn on javascript in your browser to play chess.
Posers and Puzzles

Posers and Puzzles

  1. Standard member wolfgang59
    Infidel
    23 Sep '07 19:37
    I have two standard 6-sided dice; a red one and a blue one. What is the average (mean, mode & median) difference between them.

    What about with two red dice and two blue. What is the average difference between the total of the red dice and the total of the blue dice?

    And finally for mathematicians ... N dice.
  2. Standard member HandyAndy
    Non sum qualis eram
    23 Sep '07 21:39
    Originally posted by wolfgang59
    I have two standard 6-sided dice; a red one and a blue one. What is the average (mean, mode & median) difference between them.
    1.9444...
  3. 23 Sep '07 22:35 / 1 edit
    I have only an answer when the number of dice tends to infinity.

    Let there be n red dice and n blue dice. The number on one dice has mean 3.5 and variance 35/12. So the total of the red dice has approximately normal distribution, with expected value 3.5n and variance (35/12)n. The same is true for the blue dice, and also for minus 1 times the total of the blue dice.
    The difference of two independent random variables ditributed normally has also normal distribution, so Total of red minus total of blue = total of red plus (minus 1 times total of blue) is approximately distributed normally with mean 0 and variance (35/6)n, or standard deviation sqrt(35/6)*sqrt(n).
    If a random variable x is distributed normally with mean 0 and standard deviation s, then the expected value of |x| is sqrt(2/pi)*s. So:
    for large n, the expected diffrence between the total of the n red
    dice and the total of the n blue dice is approximately sqrt(35/3pi)*sqrt(n), or about 1.927*sqrt(n).
  4. Subscriber coquette
    Already mated
    23 Sep '07 23:36
    i don't care
  5. Subscriber AThousandYoung
    It's about respect
    24 Sep '07 01:26 / 1 edit
    Originally posted by wolfgang59
    I have two standard 6-sided dice; a red one and a blue one. What is the average (mean, mode & median) difference between them.

    What about with two red dice and two blue. What is the average difference between the total of the red dice and the total of the blue dice?

    And finally for mathematicians ... N dice.
    36 possibilities.

    In 6 of them, the difference is 0.
    In 10 of them, the difference is 1.
    In 8 of them, the difference is 2.
    In 6 of them, the difference is 3.
    In 4 of them, the difference is 4.
    In 2 of them, the difference is 5.

    (6*0 + 10*1 + 8*2 + 6*3 + 4*4 + 2*5)/36 = 70/36

    Yep, 1.9444...

    N dice? Too much work for my brain right now.
  6. Standard member preachingforjesus
    Iron Pillar
    24 Sep '07 20:05 / 1 edit
    Originally posted by AThousandYoung
    36 possibilities.

    In 6 of them, the difference is 0.
    In 10 of them, the difference is 1.
    In 8 of them, the difference is 2.
    In 6 of them, the difference is 3.
    In 4 of them, the difference is 4.
    In 2 of them, the difference is 5.

    (6*0 + 10*1 + 8*2 + 6*3 + 4*4 + 2*5)/36 = 70/36

    Yep, 1.9444...

    N dice? Too much work for my brain right now.
    slight flaw in that.
    In some cases the red is greater in some the blue is greater; so you have two questions:
    What is the absolute difference?

    "ATthousandYoung" answered the mean part of this question

    In 6 of them, the difference is 0.
    In 10 of them, the difference is 1.
    In 8 of them, the difference is 2.
    In 6 of them, the difference is 3.
    In 4 of them, the difference is 4.
    In 2 of them, the difference is 5.

    Mean = 70/36 = 1.94... (last digit repeats)
    Mode = 1
    Median = 2

    What is the difference of red with respect to blue?
    In 1 of them, the difference is -5
    In 2 of them, the difference is -4
    In 3 of them, the difference is -3
    In 4 of them, the difference is -2
    In 5 of them, the difference is -1
    In 6 of them, the difference is 0.
    In 5 of them, the difference is 1.
    In 4 of them, the difference is 2.
    In 5 of them, the difference is 3.
    In 2 of them, the difference is 4.
    In 1 of them, the difference is 5.

    Mean = 0
    Median = 0
    M0de = 0

    As for two dice;
    relative difference gives the same results


    Absolut diference gives:

    In 146 cases, the diffeence is 0
    In 280 cases, the diffeence is 1
    In 250 cases, the diffeence is 2
    In 208 cases, the diffeence is 3
    In 160 cases, the diffeence is 4
    In 112 cases, the diffeence is 5
    In 70 cases, the diffeence is 6
    In 40 cases, the diffeence is 7
    In 20 cases, the diffeence is 8
    In 8 cases, the diffeence is 9
    In 2 cases, the diffeence is 10

    Mean = 3556/1296 = 889/324 = 2.74382716049...(last 9 digits repeat)
    Median = 2
    Mode = 1
  7. Subscriber coquette
    Already mated
    24 Sep '07 20:27
    okay, now i really don't care
  8. Standard member TheMaster37
    Kupikupopo!
    25 Sep '07 11:59
    Originally posted by preachingforjesus
    [b]slight flaw in that.
    In some cases the red is greater in some the blue is greater; so you have two questions:
    What is the absolute difference?

    "ATthousandYoung" answered the mean part of this question

    In 6 of them, the difference is 0.
    In 10 of them, the difference is 1.
    In 8 of them, the difference is 2.
    In 6 of them, the di ...[text shortened]... an = 3556/1296 = 889/324 = 2.74382716049...(last 9 digits repeat)
    Median = 2
    Mode = 1[/b]
    Difference is always positive.

    The difference between a and b is |a - b|.
  9. Standard member wolfgang59
    Infidel
    25 Sep '07 14:58
    Thank you .. that is what I meant as the difference. Now anyone going to tackle N dice? (Other than the excellent approximation for large values of N)??
  10. Standard member PBE6
    Bananarama
    25 Sep '07 16:21
    Originally posted by wolfgang59
    Thank you .. that is what I meant as the difference. Now anyone going to tackle N dice? (Other than the excellent approximation for large values of N)??
    How do you define the difference with N dice? Do you mean for us to first take the difference between every possible pair of dice, then average them out?
  11. Standard member wolfgang59
    Infidel
    25 Sep '07 20:51
    Originally posted by PBE6
    How do you define the difference with N dice? Do you mean for us to first take the difference between every possible pair of dice, then average them out?
    yes
    mean, mode & median

    I guess we all realise the mode is 1 for any N

    And probably (but how to prove?) that the median is 2

    But whats the mean?
  12. Standard member TheMaster37
    Kupikupopo!
    26 Sep '07 20:39
    Originally posted by wolfgang59
    yes
    mean, mode & median

    I guess we all realise the mode is 1 for any N

    And probably (but how to prove?) that the median is 2

    But whats the mean?
    Oh, I thought we were supposed to total the result of N red dice, same for the blue dice and take the diference between the two totals. My bad :p
  13. Subscriber AThousandYoung
    It's about respect
    26 Sep '07 20:39 / 1 edit
  14. Standard member Palynka
    Upward Spiral
    28 Sep '07 15:13 / 2 edits
    Originally posted by David113
    I have only an answer when the number of dice tends to infinity.

    Let there be n red dice and n blue dice. The number on one dice has mean 3.5 and variance 35/12. So the total of the red dice has approximately normal distribution, with expected value 3.5n and variance (35/12)n. The same is true for the blue dice, and also for minus 1 times the total of th ...[text shortened]... and the total of the n blue dice is approximately sqrt(35/3pi)*sqrt(n), or about 1.927*sqrt(n).
    This is false, because the variance of the sum goes to infinity, as well. So you can't use the Central Limit Theorem and say it approaches normality.
  15. Standard member Palynka
    Upward Spiral
    01 Oct '07 12:14
    Can anyone send me a PM with the answer regarding N-dice?