Re: Statistics questions


Sonja Elen Kisa wrote:
If I roll three six-sided dice (3d6) and only take the median roll (not
the mean or average), what kind of bell curve or distribution odds
would it give me for the result 1 to 6?


There are 216 possible rolls, and by symmetry the number of 1s and 6s
are the same, as are 2s and 5s, and 3s and 4s, so we only need to
consider those.

How to get a 1 (or 6)? you need to roll at least two 1s on the three
dice to have the median be 1. There are 16 ways this can happen (3 ways
to get two 1s, wih 6 choices for the other die in each, minus 2 for the
triple-counting of 1-1-1).
There are 16 ways to get a 1(6)

How to get 2 (or 5)? you need (i) exactly one 1 and at least one 2, or
(ii)no 1s and at least two 2s.
(i) 3 ways for a 1, 9 ways for each to have at least one 2 and no more
1s. total: 27
(ii) 3 ways for two 2s, each with 5 choices for the other die, less 2
for the triple-counted 2-2-2. total: 13.
There are 40 ways to get a 2(5)

216 - 16 - 16 - 40 - 40=104 remaining die rolls. By symmetry, 52 for
each of 3 and 4.

So, the probability of getting each number is (rounded to 3 decimal
places):
..074, .185, .241, .241, .185, .074

If I roll two 20-sided dice (2d20) and only look at the higher result
of the two, what is the average result I will get? What about 3d20 and
only look at the highest die? 4d20 etc. up to 16d20...?

You are looking for a number where the probability of rolling at least
one number higher is 50%.
let p=probability one die is less than or equal to n=n/20.
For M dice, we want the 1-prob(all dice <=n)=.5
Thus, we have:
1-p^M=.5, so
p^M=.5
and p=0.5^(1/M)
thus n=20*(0.5^(1/M))=20/2^(1/M)

For 2 dice, n=20/sqrt(2)=14.1..., so 14 is the most likely outcome
For 3 dice, n=20/2^(1/3)=15.87, so 16 is the most likely outcome

To determine whether to round up or down, find the probability of all
dice less than the bracketing numbers. Whichever is closer to 50% is
what you most likely want.

You can find the probabilities for up to 16 dice on your own.

.

Relevant Pages

  • Number Due
    ... in line with the predicted frequency distribution. ... It is apparant to me the dice behave in a highly structured fashion. ... Our problem as crap players is The Law of Probability, to be useful, must ...
    (rec.gambling.craps)
  • Re: Another probability "paradox".
    ... got on the blue die. ... where my chances are 1 in 6. ... After the dice have been rolled, the "probability" of two sixes is ...
    (sci.math)
  • Re: No result is likely
    ... The SINGLE ROLL probabilities tell us that the house will ... 33 probability can NEVER occur, ... so the die would just hang in mid-air forever. ... dice and make up any rules we wanted to to create a game with dice. ...
    (rec.gambling.craps)
  • No result is likely
    ... SINGLE ROLL just as well as it does for millions of rolls." ... When you roll one fair die, making sure it travels down the table and ... show, and since there are six faces, the probability of each showing ... of dice and make up any rules we wanted to to create a game with dice. ...
    (rec.gambling.craps)
  • Re: No result is likely!!!
    ... Mr Shank is using one die and when one die is used each ... result of 1 through 6 has an equal chance of showing. ... But when two dice are used, each result DOES NOT have an equal chance ... for two dice the perfect distribution changes? ...
    (rec.gambling.craps)