September 23, 2008

• Silly Math Tricks

  So, one of the common tricks in roleplaying games to provide a way of improving rolls is to do something like roll twice and take the higher of the two values. I got curious as to how exactly the probabilities fell out. I started by calculating it for rolling twice and taking the higher of the two values (using a d20 because it's a nice round number). Normal distribution, of course, is a chance of 1 out of 20 for each number. Rolling two dice and taking the higher of the two values, you of course start with 20*20 =400 possible results and the distribution turns out to be:
  

  #	Probability
  1	1
  2	3
  3	5
  4	7
  5	9
  6	11
  7	13
  8	15
  9	17
  10	19
  11	21
  12	23
  13	25
  14	27
  15	29
  16	31
  17	33
  18	35
  19	37
  20	39

  There's an obvious pattern here with the probability increasing by two with each step. So, I tried to extend it out to three dice. At this point, I stopped trying to calculate out the probabilities of the output and I quickly wrote out a table in longhand for the results of rollind a d3 three times. I got the following results:
  

  #	Probability
  1	1
  2	7
  3	19

  At this point, the lightbulb came on and I was able to generalize this out to that the probability at a given number was the same as the total number of possibilties for a sided die of that size (n³ for three dice, minus the prior numbers. So, in this case, the chance of getting 3 was 27 - 7 - 1 = 19. The chance of getting 4 would be 64 - 19 - 7 - 1 = 37. And, since the previous number consisted of a cube with the smaller numbers subtracted, this abstracts out to taking the cube of the current number and subtracting the cube of the prior number. So the change of getting a 20 by rolling 3 20's and taking the best result is 20³ - 19³ = 1141 out of 8000, a bit more than a 1 in 8 chance. This extends out to however many dice you want to add. The chance of getting a 20 by rolling 20 dice and taking the best result is about 64%, or (20²⁰ - 19²⁰) / 20²⁰. Neat, huh? ^_^ Yes, I know that I'm a geek.

  Nothing huge to report... I had a very lazy weekend. My desktop is still down (turns out it's not the power supply after all) so I'm trying to find out who's a reliable repair place in town. My fortress in Dwarf Fortress continues apace. I've had a few really weird dreams lately including some odd half-awake bits while reading documentation at work including one where I suddenly had this impression that I was taking someone's fingers and slowly twisting them until the bones splintered. No, I don't know who it was or why I was doing it. And no, I have no first-hand experience to know how bone would splinter in such a situation (I suspect the joints would go first), but in the middle of reading dry technical details, *finger splinter*! {shakes head} I wonder about my psyche sometimes. In other news, I was trying a handstand while waiting for my coffee to finish brewing and pouring and all of a sudden, I hit an island of stability where I wasn't wobbling at all. I was totally still with no strain and no need to correct my motion and it lasted for a few seconds of utter stillness before the coffee machine beeped, startling me into a graceful dismount. ^_^ Now if I can just do this in class and in the roda...

  • 6:00 pm
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