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SixHorse
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November 3rd, 2011 at 6:55:34 AM permalink
First time at this site. I was hoping someone could help me derive a classical process to calculate the expected number of dice throws with two dice to obtain all of the eleven sums. I understand the same problem with one die and the six possible outcomes, but I'm having difficulty when the outcomes are not equally likely. Thanks for any help.
FleaStiff
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November 3rd, 2011 at 7:00:38 AM permalink
The faces are equally likely, they just add up differently.
dm
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November 3rd, 2011 at 10:29:44 AM permalink
Quote: FleaStiff

The faces are equally likely, they just add up differently.

Really? You think that is the answer?
dwheatley
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November 3rd, 2011 at 11:17:06 AM permalink
Let X be a random variable (r.v.) that represents the sum of both dice.
Let X1 be r.v. for outcome of die 1
Let X2 be r.v. for outcome of die 2

P( X = 2 ) = P( X1 = 1 ) * P( X2 = 1 ) = 1/36
P( X = 3 ) = P( X1 = 1 ) * P( X2 = 2 ) + P( X1 = 2 ) * P( X2 = 1 ) = 2/36

and so forth.

E[X] = 7

Edit: I clearly misunderstood the OP
Wisdom is the quality that keeps you out of situations where you would otherwise need it
mustangsally
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November 3rd, 2011 at 1:45:28 PM permalink
removed
silly
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SixHorse
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November 6th, 2011 at 11:39:11 AM permalink
I tried to find BrunoZ and could not find the post. I did arrive at the same results in different simulation. Thanks for the info, but I would like to find BrunoZ.
Wizard
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November 6th, 2011 at 5:38:56 PM permalink
Quote: mustangsally

This was solved this past summer over at 2+2 forum by the handsome math guru BruceZ.

I used the BruceZ method and agree with his answer.

For now, I don't think I can explain his answer any better than he did, but will try for my next Ask the Wizard column. Meanwhile, the following table shows the expected number of rolls to get a 2, 2-3, 2-4, 2-5, ...

Highest Number Prob Exp wait Prob rolled before last Prob rolled last Exp rolls
2 0.027778 36.0 0.000000 1.000000 36.000000
3 0.055556 18.0 0.666667 0.333333 42.000000
4 0.083333 12.0 0.850000 0.150000 43.800000
5 0.111111 9.0 0.922222 0.077778 44.500000
6 0.138889 7.2 0.956044 0.043956 44.816484
7 0.166667 6.0 0.973646 0.026354 44.974607
8 0.138889 7.2 0.962994 0.037006 45.241049
9 0.111111 9.0 0.944827 0.055173 45.737607
10 0.083333 12.0 0.911570 0.088430 46.798765
11 0.055556 18.0 0.843824 0.156176 49.609939
12 0.027778 36.0 0.677571 0.322429 61.217385
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
SixHorse
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November 6th, 2011 at 8:50:59 PM permalink
Thanks heaps Sally - I knew it was something simple. SixHorse
MathExtremist
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November 6th, 2011 at 10:40:56 PM permalink
Quote: mustangsally

You want to know the average wait time for all 11 numbers to roll from 2 dice. Excellent math question.

This is not even a normal distribution.
The mean of 61.2 is not even close to a mode of around 35-38 and they are not close to the median of 52.

I would like to see someone here at WoV create a formula that would solve for the distribution.

It's not normal, it's Poisson. See this post for the formula (from a textbook by Sheldon Ross).
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice." -- Girolamo Cardano, 1563
mustangsally
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November 8th, 2011 at 4:52:07 PM permalink
removed
silly
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mustangsally
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June 2nd, 2018 at 9:02:04 AM permalink
Quote: mustangsally

I would like to see someone here at WoV create a formula that would solve for the distribution.

Quote: MathExtremist

It's not normal, it's Poisson. See this post for the formula (from a textbook by Sheldon Ross).

ok.
there are a few solutions for calculating the distribution (including simulation that was easiest) but no data from any of them could I find.

I used the Markov chain solution from question 29 (page 38)
in this pdf after a first/second attempt in Excel and R stalled:
http://www.madandmoonly.com/doctormatt/mathematics/dice1.pdf

mean: 61.217384763957.... (769,767,316,159/12,574,325,400)
mode: 36
median: 52

the views

pmf

cumulative

the distribution from 11 to 300 rolls can be found here
https://sites.google.com/view/krapstuff/dice/2d6-stuff

it is in a Google sheet
direct link: https://goo.gl/SKfUZB
after using the PariGp code in a Dell i7 layptop
*****
a 2048x2048 transition matrix was used with
no harm to any math

that dice pdf is fun to read over (helps if you really like dice too)
he (Matthew M. Conroy)
did (and still is doing) a super job on it! Well done!!

Sally
Last edited by: beachbumbabs on Jun 4, 2018
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