Math Is Fun Forum

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Probability Question

Two numbers a and b are randomly drawn  from the unit interval {1,2,3,4,5,6,...,n} with replacement. Then 3 more numbers c and d and e are randomly drawn with replacement from the same unit interval . What is the limit of the probability that c+d+e < a+b as n approaches ∞ . Solved this but my method is gruesome.

Answer

Last edited by bobbym (2009-05-21 08:03:40)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Probability Question

The answer has to depend on n.

For example, if n=1 then the inequality can never hold, so p=0.
If n=2 then the only way it can work is with 1+1+1 < 2+2, so p = 1/32.
I would guess that p grows as n grows, and if so it will converge to some limit.

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Why did the vector cross the road?
It wanted to be normal.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability Question

Hi mathsyperson;

  True, the answer does converge to a limit as n gets larger. I'll edit the question to be more precise.

Last edited by bobbym (2009-05-21 08:04:18)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

jimmyR

Member

Registered: 2009-05-29

Posts: 143

Re: Probability Question

Big bobbym;

Elaina has asked me if you would post your solution to this problem? She says all your solutions are ugly so why haven't you?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability Question

Hi JimmyR;

You can tell her for me that when she joins the forum, then she gets to see all my ugly solutions. I also don't care who has trained her, D.S. or Colonel Erich Von Hitler. If she wants to butt heads ( butt heads? That can't be right!? ) with old bobbym, then let her join!

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

jimmyR

Member

Registered: 2009-05-29

Posts: 143

Re: Probability Question

Hi bobbym;

I told her that. She is madder than a wounded coyote.You were right about Colonel Hitler, that's who is training her. You know how good he is.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability Question

Yes, I do. Do you remember the MIT and JPL guys, how he thrashed them on the problem set? Beat me too! He is the best problem solver I have ever seen. The guy is freaky! I thought he was gone, you know deceased. Anyway, first DS and now him. How does she do it. I have been cleaning my answer up waiting for her. It is not so ugly anymore. I will be posting it here soon.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

jerseygirl

Member

Registered: 2010-05-29

Posts: 23

Re: Probability Question

Hello Big Bobbym.

So, when are you going to post your answer?   JimmyR is waiting!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability Question

Hello jerseygirl;

I will post it as soon as I am ready, make her sweat a little. Welcome to the forum!

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

jerseygirl

Member

Registered: 2010-05-29

Posts: 23

Re: Probability Question

Thank you Big Bobbym!   Glad to be here.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability Question

Why don't you introduce yourself in introductions. Great to be called Big bobbym.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

jerseygirl

Member

Registered: 2010-05-29

Posts: 23

Re: Probability Question

I'll think about it......

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability Question

Hi jerseygirl;

Post when you like or have someyhing to say. Do not forget my new title.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Probability Question

Do you come from New Jersey, jerseygirl?

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X'(y-Xβ)=0

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability Question

Two numbers a and b are randomly drawn  from the unit interval {1,2,3,4,5,6,...,n} with replacement. Then 3 more numbers c and d and e are randomly drawn with replacement from the same unit interval . What is the limit of the probability that c+d+e < a+b as n approaches ∞ .

Now for the long awaited solution. First we generate some empirical data:

For:

{1,2,3,4, ...,n}

Intuitively it is sort of clear that the 3rd column is approaching a limit.

We can get a lot from the above table, for instance the generating function of the sequence.

The recurrence for the sequence which we can now use to compute very large n.

With initial conditions.

Most important we have the formula for the winners for any n.

If w take the limit of f(n) as n approaches infinity we will have the answer. Remembering that the total number of ways to 3 then 2 with replacement from {1,2,3,...n} is n^5

This limit is done easily by converting it into:

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Probability Question

what is the general formula for the probability under n numbers though?

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X'(y-Xβ)=0

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability Question

For any n

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Probability Question

is it a pure empirical result?

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X'(y-Xβ)=0

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability Question

Hi George,Y;

Yes, I would say so. That is why I called the solution gruesome. I could use induction to prove the general solution but I am not sure it applies here.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

jimmyR

Member

Registered: 2009-05-29

Posts: 143

Re: Probability Question

Nice use of mathematica bobbym! See you later!

Last edited by jimmyR (2010-07-25 13:38:14)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability Question

Hi jimmyR;

Yes, there is a lot of package in that solution. It is probably good enough for the real world but not quite rigorous enough.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Sarah Rebekah 13

Guest

Re: Probability Question

Hi Bobbym.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability Question

Hi Sarah;

Good Morning!

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Sarah Rebekah 13

Guest

Re: Probability Question

Hi Good Morining to u.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability Question

How have you been?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.