Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#26 2011-08-16 17:33:04

gAr
Member
Registered: 2011-01-09
Posts: 3,482

Re: Tricky Coin Flipping Probability Problem

Hi all,

I checked George,Y's solution and it matches mine.

He actually means this:


which matches exactly!

Thanks George!

Anyway, I haven't understood completely yet. I am still trying.

Last edited by gAr (2011-08-16 17:37:05)


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#27 2011-08-16 18:22:34

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Tricky Coin Flipping Probability Problem

Hi gAr;

I think you found another way to do it.

Possibly he means this:

Left out any simplification that might hide his method. This is what I was looking for, for two days. A formula! No one had it.


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.

Offline

#28 2011-08-16 18:40:48

gAr
Member
Registered: 2011-01-09
Posts: 3,482

Re: Tricky Coin Flipping Probability Problem

Hi bobbym,

Okay.
His formula is spectacular, much less calculation's required!

I don't know stochastic calculus yet. Is stochastic calculus mostly used for fincance? I can't find out how it can be used here.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#29 2011-08-16 18:45:02

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Tricky Coin Flipping Probability Problem

Hi;

I do not agree. I favor your solution and here is why.

1) It was done using experimental ideas. This establishes the relationship between the Catalan numbers and the solution. Linking the problem to a lattice.

2) It provides a generating function, from which we might determine the moments, unimodality an asymptotic form and a recurrence.

The stochastic calculus is used for brownian motion. The Feller books covers it. But did not provide that formula.


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.

Offline

#30 2011-08-16 19:00:09

gAr
Member
Registered: 2011-01-09
Posts: 3,482

Re: Tricky Coin Flipping Probability Problem

Hi,

Hmmm, okay.

Anyway, I'm happy that there's another method.
Can't we have a simpler generating function from that?


?

Last edited by gAr (2011-08-16 19:39:54)


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#31 2011-08-16 19:53:55

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Tricky Coin Flipping Probability Problem

Hi gAr;

Can there be two of them? There is a theorem for Taylor series that says there is only one Taylor series for a given sequence.


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.

Offline

#32 2011-08-16 19:55:35

George,Y
Member
Registered: 2006-03-12
Posts: 1,379

Re: Tricky Coin Flipping Probability Problem

Hi,

The reason why I deduce only half of Prob(X=50) is that there are 101 possible outcomes for X in a symmetry fashion.

P(X=0)=P(X=100) P(X=1)=P(X=99) ... P(X=49)=P(X=51)

P(X=0)+P(X=1)+P(X=2)+...+P(X=49) = P(X=51)+...+P(X=98)+P(X=99)+P(X=100)= 1-P(X=50)

So 1/2=P(X<50)+P(X=50)/2


X'(y-Xβ)=0

Offline

#33 2011-08-16 20:05:38

George,Y
Member
Registered: 2006-03-12
Posts: 1,379

Re: Tricky Coin Flipping Probability Problem

The reason why we need to double the P(R(100)<15) to get P(R(t)<15) is this:

Set R(t) as the total gains after t trials.

The probability of R(100)-R(t)>0 or R(100)-R(t)<0 is the same  if R(t)=-15 for the first time and is nearly 0.5
(considering the case R(100)=R(t)).

-----E[ R(100)-R(t) ]=0 

when the player losses all the 15 bucks some time <100, the game seems over.
However, if we let the player continue to play until time=100, the chances of winning back some r bucks and losing r bucks more are the same in the coin flip case. That approximately constitute the whole sample space and the total conditional probability is almost 1 (neglecting the r=0 case)

So

P{R(100)<-15} = P{R(t)<-15,R(100)-R(t)<0} = P{R(t)<-15}/2


X'(y-Xβ)=0

Offline

#34 2011-08-16 21:36:52

gAr
Member
Registered: 2011-01-09
Posts: 3,482

Re: Tricky Coin Flipping Probability Problem

Hi George,Y,

I'm finding it hard to understand your reasoning.
Your random variable is "number of wins", mine's "number of dollars". Maybe that's making a difference here.
I'll try to think your way and see if it makes sense to me.


Hi bobbym,

I modified my previous g.f to a probability g.f, now it's "decent"!

We need the sum of the probabilities, so it's simply:


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#35 2011-08-17 00:13:57

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Tricky Coin Flipping Probability Problem

Hi gAr;

Very good! I am still in the testing phase for all these formulas.


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.

Offline

#36 2011-08-17 00:43:37

gAr
Member
Registered: 2011-01-09
Posts: 3,482

Re: Tricky Coin Flipping Probability Problem

Hi bobbym,

Okay!


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#37 2011-08-17 00:46:45

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Tricky Coin Flipping Probability Problem

Hi gAr;

I forgot you already did that for some smaller problems. I think this one is done.

Thanks to gAr and George,Y the OP's question is answered.


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.

Offline

#38 2011-08-17 01:03:51

gAr
Member
Registered: 2011-01-09
Posts: 3,482

Re: Tricky Coin Flipping Probability Problem

Hi,

And thanks for your simulation, which gave some confidence with the result!

Now, we do not know whether the OP will ever see this?


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#39 2011-08-17 02:55:23

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Tricky Coin Flipping Probability Problem

Not only is MathsIsFun it is also lonely. he may never see the answer.


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.

Offline

#40 2011-08-17 03:06:52

gAr
Member
Registered: 2011-01-09
Posts: 3,482

Re: Tricky Coin Flipping Probability Problem

Yes, that's true!


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#41 2011-08-17 03:09:17

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Tricky Coin Flipping Probability Problem

Hi;

Sometimes posters never return because they post on another forum and get another answer. Although not as good. But it is strange to me why some would just forget about the problem they wanted an answer to.


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.

Offline

#42 2011-08-17 03:17:57

gAr
Member
Registered: 2011-01-09
Posts: 3,482

Re: Tricky Coin Flipping Probability Problem

Appears strange to me too.
Maybe they do not note down the sites where they've asked questions.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#43 2011-08-17 03:27:30

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Tricky Coin Flipping Probability Problem

Hi;

That is an idea. Maybe they decide that mathematics is too tough and they become bored. Most likely someone else gave the wrong answer but answered first.


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.

Offline

#44 2011-08-17 03:40:17

gAr
Member
Registered: 2011-01-09
Posts: 3,482

Re: Tricky Coin Flipping Probability Problem

Hi,

I searched a bit for this question.
Doesn't appear to be posted anywhere else?

Whatever... I'll note down the result! After a few days, this thread drowns down the horizon, with the formulas!


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#45 2011-08-17 03:43:40

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Tricky Coin Flipping Probability Problem

Hi gAr;

I usually copy them. Later you will need this result. You never know when.


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.

Offline

#46 2011-08-17 04:26:28

gAr
Member
Registered: 2011-01-09
Posts: 3,482

Re: Tricky Coin Flipping Probability Problem

Hi bobbym,

Yes, that's right!


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#47 2011-08-17 14:58:31

George,Y
Member
Registered: 2006-03-12
Posts: 1,379

Re: Tricky Coin Flipping Probability Problem

gAr wrote:

Hi George,Y,

I'm finding it hard to understand your reasoning.
Your random variable is "number of wins", mine's "number of dollars". Maybe that's making a difference here.
I'll try to think your way and see if it makes sense to me.

Hi gAr
I used two variables:
One is number of wins as X in the total 100 trials.
One is the profit/loss R(t) after t times of trial.

Therefore R(100) = X-(100-X) = 2X-100, which is the link between the two.


X'(y-Xβ)=0

Offline

#48 2011-08-17 16:43:52

gAr
Member
Registered: 2011-01-09
Posts: 3,482

Re: Tricky Coin Flipping Probability Problem

Hi George,Y,

From #18, I understood it like this:
We are calculating P ( losses - wins > 15 ), which is the probability of going broke.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#49 2011-08-18 04:58:25

russellhq
Member
Registered: 2011-08-13
Posts: 6

Re: Tricky Coin Flipping Probability Problem

I'm here!!  I'm trying to digest all your formulae and get it into some programme code to run the calcs.

Offline

#50 2011-08-18 05:04:32

gAr
Member
Registered: 2011-01-09
Posts: 3,482

Re: Tricky Coin Flipping Probability Problem

Hi russellhq,

Glad that you are back!


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

Board footer

Powered by FluxBB