Tricky Coin Flipping Probability Problem

bobbym · 2011-08-18 05:05:40

Hi;

Lots of languages will handle that.

russellhq · 2011-08-18 05:22:28

bobbym wrote:

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.

I think I understand it, but I am not sure what you are doing here:

How do I calculate this in Excel?

gAr · 2011-08-18 05:34:10

Hi,

Something like this:

BINOMDIST(50,100,.5,0)

I checked in libreoffice, maybe the same in ms excel.

edit: Sorry, this is a probability distribution! See next post.

Last edited by gAr (2011-08-18 05:44:51)

bobbym · 2011-08-18 05:37:37

That is called an NCR or a binomial. It means 100 choose 50. There will be something in excel to do that. Search around for keywords binomial, choose, ncr, combinations.

As a test function when you have found the correct one it will get
an answer of 1.008913445455643 x 10^29.

russellhq · 2011-08-18 21:33:02

Now the hard work has been done I wonder if we can expand to a general formula:

If we have a number of events, n, with each event having a probability of success p, and the maximum deviation we want is d. What would our formula look like (I've had a go but not so sure)?

Last edited by russellhq (2011-08-18 21:34:07)

bobbym · 2011-08-21 00:24:39

Hi;

I was able to get the problem down to a 60 x 60 markov chain.
Using that I could get a closed form for this particular problem.

You must enter odd numbers because it is impossible to go broke on
an even number.

gAr · 2011-08-21 04:07:28

Hi bobbym,

That's a nice form.

Where can I read about how to get a closed form from a markov chain?

bobbym · 2011-08-21 05:27:38

Hi gAr;

Nowhere actually but I will explain it as best as I can as soon
as I get something together.

gAr · 2011-08-21 07:42:51

Hi bobbym,

Okay, thanks.

I thought of a recurrence

We may get some insight on the generating function for each dollar from this?

bobbym · 2011-08-21 08:59:46

Hi gAr;

I did not know you were around. Thanks for the recurrence. Unfortunately
Wolfram Alpha will not run the command you need to get that answer.
I do not know of another way or why it is having a problem.

gAr · 2011-08-21 15:31:26

Hi bobbym,

I don't know of a command either for 2 variable recurrence.
I checked that recurrence for small values using a 30x30 table, works fine.

I'll see whether I can get a g.f for that.

bobbym · 2011-08-21 15:49:22

Hi gAr;

Okay, I have been working with shrinking the markov chain down to a small size.
I got it down to 32 x 32 which is much better.

gAr · 2011-08-21 15:53:07

Hi bobbym,

Oh, that's great!
I'd like to see that.

bobbym · 2011-08-21 16:06:30

Hi gAr;

By observing that once you reach + 58 (which will take 43 or more tosses).
There will be 57 or less tosses left. So, upon reaching 58+ there is no way
that you can now go broke.
Therefore we call 58+ an absorbing state because computing
them in no way can effect the prob of going broke.

We can shorten the matrix down further by taking 2 throws
at a time.

For every two tosses.

Raising A to fiftieth power produces the correct answer in
the first row and last column.

gAr · 2011-08-21 16:16:51

Hi,

Okay, I'll see.

And for that recurrence, I think find a generating function may not be easy.
There is an extra condition, we must leave out u[t][0] from adding to the next toss. Otherwise, it's like playing even after going broke.

bobbym · 2011-08-21 16:17:43

Hi gAr;

I am being called away, be back in a couple.

gAr · 2011-08-21 16:20:36

Hi bobbym,

Okay, see you later.

bobbym · 2011-08-21 18:13:13

Hi;

What do you use for the initial conditions?

gAr · 2011-08-21 19:27:16

Hi,

The initial condition is

Maybe we can avoid u(t,0) in the table, since it can be calculated from the other probabilities in that row.
Hence,

bobbym · 2011-08-21 21:09:51

Hi gAr;

Okay, thanks.

Math Is Fun Forum

#51 2011-08-18 05:05:40

Re: Tricky Coin Flipping Probability Problem

#52 2011-08-18 05:22:28

Re: Tricky Coin Flipping Probability Problem

#53 2011-08-18 05:34:10

Re: Tricky Coin Flipping Probability Problem

#54 2011-08-18 05:37:37

Re: Tricky Coin Flipping Probability Problem

#55 2011-08-18 21:33:02

Re: Tricky Coin Flipping Probability Problem

#56 2011-08-21 00:24:39

Re: Tricky Coin Flipping Probability Problem

#57 2011-08-21 04:07:28

Re: Tricky Coin Flipping Probability Problem

#58 2011-08-21 05:27:38

Re: Tricky Coin Flipping Probability Problem

#59 2011-08-21 07:42:51

Re: Tricky Coin Flipping Probability Problem

#60 2011-08-21 08:59:46

Re: Tricky Coin Flipping Probability Problem

#61 2011-08-21 15:31:26

Re: Tricky Coin Flipping Probability Problem

#62 2011-08-21 15:49:22

Re: Tricky Coin Flipping Probability Problem

#63 2011-08-21 15:53:07

Re: Tricky Coin Flipping Probability Problem

#64 2011-08-21 16:06:30

Re: Tricky Coin Flipping Probability Problem

#65 2011-08-21 16:16:51

Re: Tricky Coin Flipping Probability Problem

#66 2011-08-21 16:17:43

Re: Tricky Coin Flipping Probability Problem

#67 2011-08-21 16:20:36

Re: Tricky Coin Flipping Probability Problem

#68 2011-08-21 18:13:13

Re: Tricky Coin Flipping Probability Problem

#69 2011-08-21 19:27:16

Re: Tricky Coin Flipping Probability Problem

#70 2011-08-21 21:09:51

Re: Tricky Coin Flipping Probability Problem

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