The new dice pdf is out!

bobbym · 2013-01-23 02:37:08

Hi;

Matt Conroy's dice.pdf is out and problem #10 belongs to me! Seems that computational math is spreading!

http://www.madandmoonly.com/doctormatt/ … /dice1.pdf

The pdf is filled with interesting solutions and if you are a lover of generating functions, recurrences, Markov chains and computation, all the things I babble about in Computer Math then you will love the way a real artist uses them.

Thanks Matt!

Now, about that solution to #10...

Rumpole once sarcastically said that he always found that any knowledge of the law is a severe disadvantage. Likewise, knowledge of mathematics can ruin you when you come across a problem like that. Two things hindered me in solving that problem, lack of courage, I could not bring myself to state it as a 730 state Markov chain and a stubborn reliance on using the notation and tools of linear algebra to deal with it!

anonimnystefy · 2013-02-16 07:27:56

Hi bobbym

My answer to the last question in problem #10 regarding the case in which we want a number to appear 3 times before stopping:

First, we need to find the probability P(X=i) that the first number that appears 3 times is in the i-th postition:

For the first number that appears 3 times to be in the i-th position, it must also appear exactly twice in the previous i-1 places, and all other number can appear at most 2 time. This means, also, that, by the Pigeon hole principle, i can take values between 3 and 13, only.

Now, the number of ways we can choose two numbers from the first i-1 is

, and the probability that they are the same as the i-th number is .

The rest of the numbers can appear at most 2 times, so we can use the GF

to model them. Because the total number of other numbers in the first i-1 is i-3, and because there are only 5 possible choices for those i-3 numbers, we will model this with the GF , whose x^(i-3) term's coefficient is equal to the number of ways of distributing the i-3 numbers among the five possibilities. But this only gives us the possible distribution of those i-3 numbers, without caring for theor order in the throw sequence. To account for that we need a factor of (i-3)!. So, finally, the number of ways the i-3 numbers can be distributed is . Also, the probability that those numbers will have the values assigned to them is

Combining this with the previous results gives us the probability of

.

To get the expected value, we use the standar formula E(X)=∑ i*P(X=i):

.

Now we can use a CAS to find the value of this sum, or to get a closed form.

The following code in Maxima does that:

sum(i*(i-3)!/(6^(i-1))*binomial(i-1,2)*coeff(ratexpand((1+x+x^2/2)^5),x^(i-3)),i,3,13);

4084571/559872

Which gives us the final answer of

.

The result can be generalized from 3 to k appearances before stopping as follows:

In Maxima, we could implement this as:

f(k):=sum(i*(i-k)!/(6^(i-1))*binomial(i-1,k-1)*coeff(ratexpand((sum(x^j/j!,j,0,k-1))^5),x^(i-k)),i,k,6*k-5);

Last edited by anonimnystefy (2013-03-20 08:45:50)

bobbym · 2013-02-16 09:41:05

Hi anonimnystefy;

A very fine result, well done.

anonimnystefy · 2013-02-16 09:49:52

Thanks!

bobbym · 2013-02-16 10:22:52

I have a little better understanding of it now and it has worked for all the examples given and for a few simulations of numbers that are bigger than 2 and 3.

anonimnystefy · 2013-02-16 10:25:36

That is great! I will wait for him to answer my email now.

In the meantime, we can work on a better form for that, or an asymptotic form.

Last edited by anonimnystefy (2013-02-16 10:25:44)

bobbym · 2013-02-16 10:32:12

There are one or two spots that might be simplified or maybe not. I do not yet know about an asymptotic form. It would interesting to be able to answer at least approximately for say 1000000 in a row.

Last edited by bobbym (2013-02-17 01:06:48)

anonimnystefy · 2013-02-16 11:11:28

It seems that the inner sum has a closed form in terms of the Gamma and the upper incomplete Gamma function.

bobbym · 2013-02-16 11:12:29

Yes, there is a closed form for much of that.

anonimnystefy · 2013-02-16 12:24:06

That sum is troubling me. Is there anyway to get rid of it?

bobbym · 2013-02-16 12:39:53

Which sum are we talking about?

anonimnystefy · 2013-02-16 12:49:03

The inner sum.

bobbym · 2013-02-16 13:45:03

That is for the gf, you do not have to use that approach.

anonimnystefy · 2013-02-16 13:47:14

I don't understand you.

bobbym · 2013-02-16 13:54:03

There was another way to generate T[n,k], it used a series.

anonimnystefy · 2013-02-16 13:59:50

Yes, but it cannot be used for the generalized case.

By the way, Matt answere asking what my method was. I answered him and told him I could send him the full proof and that I have found a generalization.

bobbym · 2013-02-16 14:05:40

Hi;

Why not send him what you posted?

anonimnystefy · 2013-02-16 21:28:59

I will if he says he's interested.in seeing it.

bobbym · 2013-02-16 22:56:19

Hmmm, he did ask for your method.

anonimnystefy · 2013-02-16 23:58:52

Yes. He asked for the method, but not the whole solution.

By the way, you really should not lie.

bobbym · 2013-02-17 00:04:49

Hi;

About what? Under "Thanks" you did not find my name? One of them is me!

Last edited by bobbym (2013-02-17 00:14:32)

anonimnystefy · 2013-02-17 00:19:37

Lie about this:

bobbym wrote:

Hi gAr;
I am sorry, but on this one Mr anonimnystefy has left me dumber than a burned out stump. Apparently his method has fried my brain. I am so confused I can not solve 3 + x = 9. Can you show me what you want done? You can use codecogs for the latexing.

bobbym · 2013-02-17 00:21:57

Could be that I have just got up or your solution or both but I am not particularly sharp right now.

anonimnystefy · 2013-02-17 00:23:13

Maybe it's because of your back.

Have you got an idea for clearing up the formula up there.

bobbym · 2013-02-17 00:24:11

I have been working hard on it but unfortunately no.

Math Is Fun Forum

#1 2013-01-23 02:37:08

The new dice pdf is out!

#2 2013-02-16 07:27:56

Re: The new dice pdf is out!

#3 2013-02-16 09:41:05

Re: The new dice pdf is out!

#4 2013-02-16 09:49:52

Re: The new dice pdf is out!

#5 2013-02-16 10:22:52

Re: The new dice pdf is out!

#6 2013-02-16 10:25:36

Re: The new dice pdf is out!

#7 2013-02-16 10:32:12

Re: The new dice pdf is out!

#8 2013-02-16 11:11:28

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#9 2013-02-16 11:12:29

Re: The new dice pdf is out!

#10 2013-02-16 12:24:06

Re: The new dice pdf is out!

#11 2013-02-16 12:39:53

Re: The new dice pdf is out!

#12 2013-02-16 12:49:03

Re: The new dice pdf is out!

#13 2013-02-16 13:45:03

Re: The new dice pdf is out!

#14 2013-02-16 13:47:14

Re: The new dice pdf is out!

#15 2013-02-16 13:54:03

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#16 2013-02-16 13:59:50

Re: The new dice pdf is out!

#17 2013-02-16 14:05:40

Re: The new dice pdf is out!

#18 2013-02-16 21:28:59

Re: The new dice pdf is out!

#19 2013-02-16 22:56:19

Re: The new dice pdf is out!

#20 2013-02-16 23:58:52

Re: The new dice pdf is out!

#21 2013-02-17 00:04:49

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#22 2013-02-17 00:19:37

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#23 2013-02-17 00:21:57

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#24 2013-02-17 00:23:13

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#25 2013-02-17 00:24:11

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