Expected Number

November 3rd, 2011 at 6:55:34 AM permalink
SixHorse Member since: Nov 3, 2011 Threads: 1 Posts: 3	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.

November 3rd, 2011 at 7:00:38 AM permalink
FleaStiff Member since: Oct 19, 2009 Threads: 75 Posts: 4829	The faces are equally likely, they just add up differently.

November 3rd, 2011 at 10:29:44 AM permalink
dm Member since: Apr 29, 2010 Threads: 14 Posts: 699	Quote: FleaStiff The faces are equally likely, they just add up differently. Really? You think that is the answer?

November 3rd, 2011 at 11:17:06 AM permalink

dwheatley

Member since: Nov 16, 2009
Threads: 10
Posts: 550

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

November 3rd, 2011 at 1:45:28 PM permalink

mustangsally

Member since: Mar 29, 2011
Threads: 5
Posts: 170

Quote: SixHorse
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.

You want to know the average wait time for all 11 numbers to roll from 2 dice. Excellent math question.
This was solved this past summer over at 2+2 forum by the handsome math guru BruceZ.

Dice rolling question
Some wild math.
Hope this helps.

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.
But, A Markov Chain may be the easiest way to go.

I can solve it only by simulation only.


grouped data
items:                  1,000,000   

minimum value:               11.00
first quartile:              36.00
median:                      52.00
third quartile:              76.00
maximum value:              531.00

mean value:                  61.21
midrange:                   271.00

range:                      520.00
interquartile range:         40.00
mean abs deviation:          26.53

sample variance (n):       1296.43
sample variance (n-1):     1296.43
sample std dev (n):          36.01
sample std dev (n-1):        36.01

Sorry I forgot the axis labels. Should be easy to figure out.
x-axis is the number of trials for a complete set of all 11 numbers.

Some sim data

wait	freq	rf	cf
11	25	0.000025	0.000025
12	114	0.000114	0.000139
13	355	0.000355	0.000494
14	866	0.000866	0.00136
15	1419	0.001419	0.002779
16	2241	0.002241	0.00502
17	3325	0.003325	0.008345
18	4342	0.004342	0.012687
19	5750	0.00575	0.018437
20	6899	0.006899	0.025336
21	8015	0.008015	0.033351
22	9356	0.009356	0.042707
23	10406	0.010406	0.053113
24	11604	0.011604	0.064717
25	12588	0.012588	0.077305
26	13636	0.013636	0.090941
27	14183	0.014183	0.105124
28	15405	0.015405	0.120529
29	16079	0.016079	0.136608
30	16405	0.016405	0.153013
31	16655	0.016655	0.169668
32	16827	0.016827	0.186495
33	16990	0.01699	0.203485
34	17304	0.017304	0.220789
35	17570	0.01757	0.238359
36	17458	0.017458	0.255817
37	17295	0.017295	0.273112
38	17571	0.017571	0.290683
39	17505	0.017505	0.308188
40	17153	0.017153	0.325341
41	16879	0.016879	0.34222
42	16780	0.01678	0.359
43	16700	0.0167	0.3757
44	16157	0.016157	0.391857
45	15918	0.015918	0.407775
46	15937	0.015937	0.423712
47	15524	0.015524	0.439236
48	14980	0.01498	0.454216
49	14839	0.014839	0.469055
50	14475	0.014475	0.48353
51	14067	0.014067	0.497597
52	13903	0.013903	0.5115
53	13633	0.013633	0.525133
54	13313	0.013313	0.538446
55	12869	0.012869	0.551315
56	12547	0.012547	0.563862
57	12278	0.012278	0.57614
58	11907	0.011907	0.588047
59	11450	0.01145	0.599497
60	11286	0.011286	0.610783
61	10956	0.010956	0.621739
62	10886	0.010886	0.632625
63	10578	0.010578	0.643203
64	10109	0.010109	0.653312
65	9803	0.009803	0.663115
66	9477	0.009477	0.672592
67	9242	0.009242	0.681834
68	8868	0.008868	0.690702
69	8827	0.008827	0.699529
70	8447	0.008447	0.707976
71	8359	0.008359	0.716335
72	8130	0.00813	0.724465
73	7815	0.007815	0.73228
74	7619	0.007619	0.739899
75	7647	0.007647	0.747546
76	7425	0.007425	0.754971
77	6972	0.006972	0.761943
78	6795	0.006795	0.768738
79	6628	0.006628	0.775366
80	6382	0.006382	0.781748
81	6199	0.006199	0.787947
82	6070	0.00607	0.794017
83	5769	0.005769	0.799786
84	5755	0.005755	0.805541
85	5552	0.005552	0.811093
86	5515	0.005515	0.816608
87	5308	0.005308	0.821916
88	5268	0.005268	0.827184
89	4925	0.004925	0.832109
90	4805	0.004805	0.836914
91	4769	0.004769	0.841683
92	4390	0.00439	0.846073
93	4178	0.004178	0.850251
94	4264	0.004264	0.854515
95	4067	0.004067	0.858582
96	3953	0.003953	0.862535
97	3806	0.003806	0.866341
98	3722	0.003722	0.870063
99	3670	0.00367	0.873733
100	3566	0.003566	0.877299
101	3446	0.003446	0.880745
102	3392	0.003392	0.884137
103	3159	0.003159	0.887296
104	3131	0.003131	0.890427
105	3093	0.003093	0.89352
106	3014	0.003014	0.896534
107	2918	0.002918	0.899452
108	2842	0.002842	0.902294

I Heart Vi Hart

November 6th, 2011 at 11:39:11 AM permalink
SixHorse Member since: Nov 3, 2011 Threads: 1 Posts: 3	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.

November 6th, 2011 at 5:38:56 PM permalink

Wizard

Administrator
Member since: Oct 14, 2009
Threads: 313
Posts: 6784

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

It's not whether you win or lose; it's whether or not you had a good bet.

November 6th, 2011 at 8:50:59 PM permalink
SixHorse Member since: Nov 3, 2011 Threads: 1 Posts: 3	Thanks heaps Sally - I knew it was something simple. SixHorse

November 6th, 2011 at 10:40:56 PM permalink

MathExtremist

Member since: Aug 31, 2010
Threads: 46
Posts: 2521

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

November 8th, 2011 at 4:52:07 PM permalink
mustangsally Member since: Mar 29, 2011 Threads: 5 Posts: 170	Quote: MathExtremist It's not normal, it's Poisson. See this post for the formula (from a textbook by Sheldon Ross). Thank you kind Sir. Just seeing BruceZ, The Wizard and MathExtremist all on the same page, the start of an awesome fantasy... right girls...math speaking that is. I Heart Vi Hart

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