What do you think?

bobbym · 2011-03-07 00:19:52

Yes, it agrees with one of the 5 answers given in post # 623. Also the pdf has a worked out example. I am checking it in another way.

It might be that the markov chain is missing some.

gAr · 2011-03-07 00:37:45

Ok, I'll check again.

bobbym · 2011-03-07 01:03:24

Hi gAr;

See what you think of this:

http://forum.tradingblox.com/download.php?id=2374.

gAr · 2011-03-07 01:22:54

Hi bobbym,

It's confusing me. Both seem to be correct.
Let me see if I can get something with simulation.
Did you try any other way?

bobbym · 2011-03-07 01:29:47

Yes, I tried a computer approach that would settle the question. I could only come up with a lower bound of .1712. This is not good enough, either answer could be right.

gAr · 2011-03-07 01:45:32

I did some logical mistake in my code, trying to debug it!

But it appeals more naturally to an absorbing Markov chain, what do you say? Or maybe I'm biased!

bobbym · 2011-03-07 01:49:00

Hi gAr;

I am not sure. I am just looking for any answer.

gAr · 2011-03-07 01:55:40

Hi bobbym,

Ok, I'll get back if I have something.

bobbym · 2011-03-07 02:33:32

Hi gAr;

I am getting about .186 for a simulation of 100,000 suggesting that he is correct. The trouble with the markov chain is it does not seem to have any way to determine which came first.

gAr · 2011-03-07 02:47:18

Hi bobbym,

Ok, I'm trying to learn more about the art of simulation.
I'll also try to know more about the markov chain.

bobbym · 2011-03-07 02:50:47

Hi gAr;

No matter how much you know there is always some problem that is too much. I do not have a good answer either.

gAr · 2011-03-07 03:06:13

Hi bobbym,

Got it!
The probability of absorption:

We must not go to the start state once it gets off it.
Instead, it must transition to 1T if it's in 1H or 2H and to 1H if it's in 1T,2T,3T or 4T.

Last edited by gAr (2011-03-07 03:06:59)

bobbym · 2011-03-07 03:18:03

Hi gAr;

I see what you did. That is the right answer! I did not even look at the Markov chain idea here because I just did not see how the matrix knows when the TTTTT came before the HHH. Looks like it does though, amazing. Very good!

gAr · 2011-03-07 03:44:33

Hi bobbym,

It will know about it when we take two absorbing states. Because, once it gets to the absorbing state it won't go to any other state, by the definition. Question is then only about the probability of getting into one of the absorbing states.
The main thing here is to be careful with the transitions.
I was trying initially with only one absorbing state, and I learned that alone can't do.

Last edited by gAr (2011-03-07 03:56:03)

bobbym · 2011-03-07 04:11:30

Hi gAr;

Cool thing is that using the Markov chain we get the expected time too. It is 217 / 19 throws from the start state.

gAr · 2011-03-07 04:26:20

Hi bobbym,

Yeah, only thing is that the matrix should not become too big. Then it becomes difficult to spot the mistakes.
But overall, markov chain is a cool thing!

Howardroark · 2011-03-07 04:58:36

Hi bobbym,

bobbym wrote:

New problem:
D wants to use his computer to flip a coin. So he puts a coin on top of the keyboard and goes to bed. The computer has trouble picking up the coin so D decides to build robotic arms for it. Someone tells D about programming and he take the 14 minute course in C++.After he graduates ( 3rd in his class ) he fires up his new C++ compiler and simulates a billion coin toss games. He walks up to our distinguished panel and says, " Do you guys know that the chance of getting 5 tails before you get 3 heads is 1 / 4?"
A says) Hmmm, I will have to think about that for a couple of years.
2 years later:
I got it: Let g be a group and G be a metric. If h(x) is meromorphic to... 5000 lines later. You are right D!
B says) After trying to read the 5000 lines, "That is not right. I do not know about that meromorphic jazz but the correct probability is 7 / 38."
C says) B, it is definitely 1 / 256. You just multiply ( 1 / 32 ) by ( 1 / 8 ).
D says) My code is correct. It is 1 / 4 =.25
E says) Are you sure of that 2, after all it is an approximation of an approximation. That 2 could be 1.98.

bobbym · 2011-03-07 09:30:12

Hi Howardroark;

Thanks to both of you for your solutions.

gAr · 2011-03-07 15:23:31

Hi bobbym and Howardroark,

We are not given the total number of tosses.
If we need TTTTTHHH, answer may not be 1/256.

The expected number of tosses I get is E=262
So answer may be:
p= 1/E = 1/262

Last edited by gAr (2011-03-07 15:26:06)

bobbym · 2011-03-07 19:09:09

Hi gAr;

The expected number and probability sometimes are a little strange. Remember the (3,3) and (3,4)? Their probability of happening is the same but their expected times are not.

When I posed the question I did not lock out that answer. I liked 7 / 38 also but if you look at it as meaning this:

P(TTTTTHHH in exactly 8 throws) = 1 / 256

The question does not state that approach is false. If I would have said someone flips a coin until he gets TTTTT without a run of HHH and then he stops when gets that TTTTT.

gAr · 2011-03-07 19:22:11

Hi bobbym,

Yes, I'll remember that forever!
Anyway, if it's exactly 8 throws it's an easy problem. For any number of throws, the answer isn't obvious. Can you check that too?

bobbym · 2011-03-07 19:25:02

If the game terminates on TTTTT then that answer is incorrect.

Can you check that too?

What can I check for you?

gAr · 2011-03-07 19:28:03

Hi,

I meant probability of the sequence TTTTTHHH, for any number of throws.

bobbym · 2011-03-07 19:50:55

Hi gAr;

Would that not be 1. As n approaches infinity it becomes a certainty that TTTTTHHH shows up. Or did you want the expected value?

gAr · 2011-03-07 19:59:11

Ah, yes! it would be 1.

I read about a theorem which mentions something like E = 1/p. Can we do the same by finding E, and write p=1/E? What does this represent?
I'm a little confused about this.

Math Is Fun Forum

#626 2011-03-07 00:19:52

Re: What do you think?

#627 2011-03-07 00:37:45

Re: What do you think?

#628 2011-03-07 01:03:24

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#629 2011-03-07 01:22:54

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#630 2011-03-07 01:29:47

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#631 2011-03-07 01:45:32

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#632 2011-03-07 01:49:00

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#634 2011-03-07 02:33:32

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#635 2011-03-07 02:47:18

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#639 2011-03-07 03:44:33

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#640 2011-03-07 04:11:30

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#643 2011-03-07 09:30:12

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#645 2011-03-07 19:09:09

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#646 2011-03-07 19:22:11

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#647 2011-03-07 19:25:02

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