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NowTheSerpent
Joined: Sep 30, 2011
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November 9th, 2011 at 7:52:09 AM permalink
Quote: DJTeddyBear

I was afraid to admit that I have no idea what he's even referring to when he says "as everyone knows...."

I'm sure he's attempting a bit of humor. And stochastic means purely probabilistic, with no intelligence controlling the outcome.
NowTheSerpent
Joined: Sep 30, 2011
• Threads: 15
• Posts: 417
November 9th, 2011 at 7:53:48 AM permalink
Quote: Wizard

You're right, thanks for the correction. The column should be correct now.

I could count the number of girls on this forum with one hand, so don't expect one of them to answer. Personally, I think that is the mathematician from the show Numbers.

Doctor Charles Epps
7craps
Joined: Jan 23, 2010
• Threads: 18
• Posts: 1977
November 9th, 2011 at 1:53:36 PM permalink
Quote: CrystalMath

I actually don't have it any more and it would be pretty time consuming to recreate. I used a Markov Chain instead of a matrix though. They essentially do the samem thing but the chain will use less space in Excel.

I figured you might not still have your work.
You use Excel. That is a lot of work for matrices.
There are free softs (Winmat at Peanut soft) Winmat
and online like
Matrix Algebra Tool that are even easier and faster.

Is not the matrix below correct except I do not know what to place in row 9 to make this 9x9?
(A 1 in 9,9 would be for 8 or more and that is a different question.)

0.50000 0.50000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.50000 0.00000 0.50000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.50000 0.00000 0.00000 0.50000 0.00000 0.00000 0.00000 0.00000 0.00000
0.50000 0.00000 0.00000 0.00000 0.50000 0.00000 0.00000 0.00000 0.00000
0.50000 0.00000 0.00000 0.00000 0.00000 0.50000 0.00000 0.00000 0.00000
0.50000 0.00000 0.00000 0.00000 0.00000 0.00000 0.50000 0.00000 0.00000
0.50000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.50000 0.00000
0.50000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.50000
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
We would be concerned with 8,1 a tail after 7 heads correct?
Plugging in a few numbers does not arrive at your answer.

I can do the work, I just need to know how to start off.
Thanks
winsome johnny (not Win some johnny)
CrystalMath
Joined: May 10, 2011
• Threads: 8
• Posts: 1893
November 9th, 2011 at 8:13:38 PM permalink
I thought about this again since I just can't stay away from interesting questions. Anyhow, the answer I was trying to find was the probability of getting exactly one streak of 7. Because of that, we have to keep track of when the streak exceeds 7 and also we need to keep track of whether or not a streak of 7 has already been reached. In the matrix below, "streak 7,n" represents the likelihood of having reached a streak of 7 and then currently having a streak of size n. When you calculate T^100, using the following transition matrix, the probability of ending in each state will be on the top row. Then, we need to add the probabilities of streak 7; streak 7,0 through streak 7,6; and streak 7,8+. When you do, you will see that I made an error in my original answer. In my original answer, it appears as though I also added streak 7,7.

`streak 0	0.5	0.5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0streak 1	0.5	0	0.5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0streak 2	0.5	0	0	0.5	0	0	0	0	0	0	0	0	0	0	0	0	0	0streak 3	0.5	0	0	0	0.5	0	0	0	0	0	0	0	0	0	0	0	0	0streak 4	0.5	0	0	0	0	0.5	0	0	0	0	0	0	0	0	0	0	0	0streak 5	0.5	0	0	0	0	0	0.5	0	0	0	0	0	0	0	0	0	0	0streak 6	0.5	0	0	0	0	0	0	0.5	0	0	0	0	0	0	0	0	0	0streak 7	0	0	0	0	0	0	0	0	0.5	0.5	0	0	0	0	0	0	0	0streak 8+	0.5	0	0	0	0	0	0	0	0.5	0	0	0	0	0	0	0	0	0streak 7,0	0	0	0	0	0	0	0	0	0	0.5	0.5	0	0	0	0	0	0	0streak 7,1	0	0	0	0	0	0	0	0	0	0.5	0	0.5	0	0	0	0	0	0streak 7,2	0	0	0	0	0	0	0	0	0	0.5	0	0	0.5	0	0	0	0	0streak 7,3	0	0	0	0	0	0	0	0	0	0.5	0	0	0	0.5	0	0	0	0streak 7,4	0	0	0	0	0	0	0	0	0	0.5	0	0	0	0	0.5	0	0	0streak 7,5	0	0	0	0	0	0	0	0	0	0.5	0	0	0	0	0	0.5	0	0streak 7,6	0	0	0	0	0	0	0	0	0	0.5	0	0	0	0	0	0	0.5	0streak 7,7	0	0	0	0	0	0	0	0	0	0.5	0	0	0	0	0	0	0	0.5streak 7,8+	0	0	0	0	0	0	0	0	0	0.5	0	0	0	0	0	0	0	0.5`
I heart Crystal Math.
7craps
Joined: Jan 23, 2010
• Threads: 18
• Posts: 1977
November 9th, 2011 at 9:05:53 PM permalink
Quote: CrystalMath

I thought about this again since I just can't stay away from interesting questions. Anyhow, the answer I was trying to find was the probability of getting exactly one streak of 7.

OK.
Excellent work but I now think you are answering a different question.

Now this seems confusing.
Starting over...

From the Wizard:
"The probability of getting "one or more" streaks of exactly seven heads in a row out of 100 coin tosses is 17.29%"
I get the same answer from computer sims.

"The probability of getting one or more streaks of "at least" seven heads in a row out of 100 coin tosses is 31.7520%"
The Wizard has the correct solution of 31.7520% but had the matrix not square in his article. He did correct it.

June 16, you wrote, "Using a Markov chain, I calculate the probability of exactly one 7 coin streak to be 0.172920892."
So then you must have added all the values?
I must have misread that statement when I first read it.

But is sure looks like the theoretical answer that matches the answer the Wizard gave for "one or more". The Wizard has never shown his work on this so I must assume he also ran a computer simulation.

I will work with your matrix and see if I can duplicate your first answer by adding all the values. It should work.
Thank you for your efforts.
Excellent work as always.

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Added: Excellent. I came up with 0.1729209 and I am sure if I added the extra places the results will match perfectly to your answer.

Worked exactly as you described and I now understand the process.

I did not use Excel to calculate T^100, I used Winmat (free Peanut soft) and it took just a few seconds to paste in the matrix once I had it set up in Excel.

I have seen others use inclusion-exclusion for these type of problems and they are really messy to work with.

I hope others can understand your work as well, if they want to of course.

Again, Thanks very much.

Added Nov 10.
Here is my T^100
`streak 0	streak 1	streak 2	streak 3	streak 4	streak 5	streak 6	streak 7	streak 8+	streak 7,0	streak 7,1	streak 7,2	streak 7,3	streak 7,4	streak 7,5	streak 7,6	streak 7,7	streak 7,8+0.414358508	0.207589542	0.104000322	0.052103139	0.026103161	0.013077427	0.006551663	0.003282319	0.003295345	0.085641492	0.042410458	0.020999678	0.010396861	0.005146839	0.002547573	0.001260837	0.000623931	0.0006109050.412720599	0.206768966	0.103589221	0.051897182	0.025999978	0.013025734	0.006525765	0.003269344	0.003282319	0.087279401	0.043231034	0.021410779	0.010602818	0.005250022	0.002599266	0.001286735	0.000636906	0.0006239310.409451255	0.205131057	0.102768644	0.051486081	0.025794021	0.012922551	0.006474071	0.003243446	0.003256318	0.090548745	0.044868943	0.022231356	0.011013919	0.005455979	0.002702449	0.001338429	0.000662804	0.0006499320.40292549	0.201861713	0.101130735	0.050665505	0.02538292	0.012716594	0.006370889	0.003191753	0.003204419	0.09707451	0.048138287	0.023869265	0.011834495	0.00586708	0.002908406	0.001441611	0.000714497	0.0007018310.389899757	0.195335948	0.097861391	0.049027596	0.024562344	0.012305493	0.006164931	0.00308857	0.003100827	0.110100243	0.054664052	0.027138609	0.013472404	0.006687656	0.003319507	0.001647569	0.00081768	0.0008054230.363899778	0.182310214	0.091335626	0.045758252	0.022924435	0.011484917	0.00575383	0.002882612	0.002894052	0.136100222	0.067689786	0.033664374	0.016741748	0.008325565	0.004140083	0.00205867	0.001023638	0.0010121980.312002596	0.156310236	0.078309893	0.039232487	0.019655091	0.009847007	0.004933254	0.002471512	0.00248132	0.187997404	0.093689764	0.046690107	0.023267513	0.011594909	0.005777993	0.002879246	0.001434738	0.001424930.208413375	0.104413054	0.052309914	0.026206753	0.013129326	0.006577663	0.003295345	0.001650935	0.001657487	0.291586625	0.145586946	0.072690086	0.036293247	0.018120674	0.009047337	0.004517155	0.002255315	0.0022487630.416002918	0.208413375	0.104413054	0.052309914	0.026206753	0.013129326	0.006577663	0.003295345	0.003308423	0.083997082	0.041586625	0.020586946	0.010190086	0.005043247	0.002495674	0.001234837	0.000610905	0.0005978270	0	0	0	0	0	0	0	0	0.5	0.25	0.125	0.0625	0.03125	0.015625	0.0078125	0.00390625	0.003906250	0	0	0	0	0	0	0	0	0.5	0.25	0.125	0.0625	0.03125	0.015625	0.0078125	0.00390625	0.003906250	0	0	0	0	0	0	0	0	0.5	0.25	0.125	0.0625	0.03125	0.015625	0.0078125	0.00390625	0.003906250	0	0	0	0	0	0	0	0	0.5	0.25	0.125	0.0625	0.03125	0.015625	0.0078125	0.00390625	0.003906250	0	0	0	0	0	0	0	0	0.5	0.25	0.125	0.0625	0.03125	0.015625	0.0078125	0.00390625	0.003906250	0	0	0	0	0	0	0	0	0.5	0.25	0.125	0.0625	0.03125	0.015625	0.0078125	0.00390625	0.003906250	0	0	0	0	0	0	0	0	0.5	0.25	0.125	0.0625	0.03125	0.015625	0.0078125	0.00390625	0.003906250	0	0	0	0	0	0	0	0	0.5	0.25	0.125	0.0625	0.03125	0.015625	0.0078125	0.00390625	0.003906250	0	0	0	0	0	0	0	0	0.5	0.25	0.125	0.0625	0.03125	0.015625	0.0078125	0.00390625	0.00390625`

0.1729208930 is my final answer
winsome johnny (not Win some johnny)
CrystalMath
Joined: May 10, 2011
• Threads: 8
• Posts: 1893
November 10th, 2011 at 8:29:12 AM permalink
After re-reading the entire post, I see that we were trying to confirm the Wizard's assertion that the probability of getting "one or more" streaks of length 7 is 0.1729. In my original answer, I said "exactly one" streak of length 7, but my answer was for "one or more."

So, it looks like we agree.

Thanks for the Winmat suggestion. I'll need to look into that. Also, I need to finally try to learn about inclusion/exclusion.
I heart Crystal Math.
7craps
Joined: Jan 23, 2010
• Threads: 18
• Posts: 1977
November 10th, 2011 at 8:44:28 AM permalink
Quote: CrystalMath

Thanks for the Winmat suggestion. I'll need to look into that. Also, I need to finally try to learn about inclusion/exclusion.

Thank you. You are very welcome.

inclusion-exclusion is a great tool to have. It can get very messy when using many terms.

a good example is here:
askthewizard/278
4th question down
"What is the expected number of rolls of two dice for every total from 2 to 12 to occur at least once?"

Also BruceZ over at 2+2 really does a great job on explaining inclusion-exclusion
The Wizard explains it also in the article.
winsome johnny (not Win some johnny)
7craps
Joined: Jan 23, 2010
• Threads: 18
• Posts: 1977
November 10th, 2011 at 10:01:58 AM permalink
Quote: CrystalMath

Thanks for the Winmat suggestion. I'll need to look into that.

Winmat is a Windows program from Peanut software, its free, that does matrices and much more.
An excellent program as is the suite of Peanut math programs.

I suggest you try it out with the:
"I once hit six royals in single-line video poker within 5,000 hands.
In my lifetime I have played about 25 million hands. What are the odds?"
matrix that you did.
six royals in single-line video poker within 5,000 hands-ask the wizard 277

You can just copy and paste the matrix into Winmat and calculate T^25,000,000 in a split second.
(or 25,000,000-5000) I have not studied that article yet.

I have only used the Winmat program a few times but what an excellent tool to have IMO.

Thanks for all your excellent work!
winsome johnny (not Win some johnny)
mustangsally
Joined: Mar 29, 2011
• Threads: 25
• Posts: 2463
February 14th, 2012 at 11:17:03 AM permalink
Happy Valentine's Day
I am on vacation!
Yeah!

i got it too, once i got it

Excel transition matrix

recursion matrix answer

a full view of the recursion markov chain
I Heart Vi Hart
mustangsally
Joined: Mar 29, 2011
• Threads: 25
• Posts: 2463
July 25th, 2015 at 9:12:23 AM permalink
Quote: pacomartin

I am not sure if the third answer is correct.
The probability of getting one or more streaks of at least seven heads in a row out of 100 coin tosses is 31.752%
The probability of getting one or more streaks of at least eight heads in a row out of 100 coin tosses is 17.021%

The answer should be the difference 31.752%-17.021%=14.731% (which is different than either your answer or the Wizard's).

It's possible I am the one making the mistake here, but I will post anyway.

i agree with all your probabilities and say you made no mistake ;)

14.731% is the answer to this
"What is the probability that the longest run of heads is exactly 7 in 100 flips?"

in other words
it could be this sequence at the beginning
HHH HHHH T
or this some where in 100 flips
T HHH HHHH T
or this at the end
T HHH HHHH

a Markov chain can solve this too (Julybe later)
as well as a simulation (already done)
here is the distribution of the longest run in N trials from me Excel

and this paper has a solution too
"Notes on the longest run of heads"
Bret Larget
September 18, 2009

now to see (C) what i can do B4 the 6pm Angel game start time
to get them to score more runs (at least more than 2)
Mike and Albert (they LOVE pizza) get ready to go OVER!
I Heart Vi Hart

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