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How many samlples from 10 digits?
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Stig Holmquist
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Joined: 30 Apr 2005
Posts: 48

PostPosted: Tue Jul 11, 2006 3:30 pm    Post subject: How many samlples from 10 digits? Reply with quote

Suppose I draw one sample at a time with replacement from a bin with
10 balls numbered 0 to 9 and do this ten times. How many different
sets of ten can I generate disregarding order?

A few extreme sets would be e.g. ten of the same digit or a mix of
only two digits or 0123456789.

Suppose I calculate the std.dev. for each set and graph the
various frequencies. Would I get a chi-square curve with 9 df?

Stig Holmquist
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mensanator@aol.compost
science forum Guru


Joined: 24 Mar 2005
Posts: 826

PostPosted: Tue Jul 11, 2006 5:09 pm    Post subject: Re: How many samlples from 10 digits? Reply with quote

Stig Holmquist wrote:
Quote:
Suppose I draw one sample at a time with replacement from a bin with
10 balls numbered 0 to 9 and do this ten times. How many different
sets of ten can I generate disregarding order?

92378

Quote:

A few extreme sets would be e.g. ten of the same digit or a mix of
only two digits or 0123456789.

Using a generator to find combinations w/replacement, I get for

2 balls - 3 combinations
3 balls - 10 combinations
4 balls - 35 combinations
5 balls - 126 combinations

and seeing where these fall on Pascal's triangle, the answer for
m balls is C(2m-1,m).

Quote:

Suppose I calculate the std.dev. for each set and graph the
various frequencies. Would I get a chi-square curve with 9 df?

Uh...

Quote:

Stig Holmquist
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Stig Holmquist
science forum beginner


Joined: 30 Apr 2005
Posts: 48

PostPosted: Tue Jul 11, 2006 8:30 pm    Post subject: Re: How many samlples from 10 digits? Reply with quote

On 11 Jul 2006 10:09:15 -0700, "mensanator@aol.com"
<mensanator@aol.com> wrote:

Quote:
Stig Holmquist wrote:
Suppose I draw one sample at a time with replacement from a bin with
10 balls numbered 0 to 9 and do this ten times. How many different
sets of ten can I generate disregarding order?

92378

This number does not seem realistic considering that the cchances

of drawing the same digit 10 times is only 1 per 10^10

Stig Holmquist>

Quote:

A few extreme sets would be e.g. ten of the same digit or a mix of
only two digits or 0123456789.

Using a generator to find combinations w/replacement, I get for

2 balls - 3 combinations
3 balls - 10 combinations
4 balls - 35 combinations
5 balls - 126 combinations

and seeing where these fall on Pascal's triangle, the answer for
m balls is C(2m-1,m).


Suppose I calculate the std.dev. for each set and graph the
various frequencies. Would I get a chi-square curve with 9 df?

Uh...


Stig Holmquist
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mensanator@aol.compost
science forum Guru


Joined: 24 Mar 2005
Posts: 826

PostPosted: Tue Jul 11, 2006 11:16 pm    Post subject: Re: How many samlples from 10 digits? Reply with quote

Stig Holmquist wrote:
Quote:
On 11 Jul 2006 10:09:15 -0700, "mensanator@aol.com"
mensanator@aol.com> wrote:

Stig Holmquist wrote:
Suppose I draw one sample at a time with replacement from a bin with
10 balls numbered 0 to 9 and do this ten times. How many different
sets of ten can I generate disregarding order?

92378

This number does not seem realistic considering that the cchances
of drawing the same digit 10 times is only 1 per 10^10

Why do you think that?

Suppose the urn contained only two balls 0 & 1
and you selected 32 with replacement. There are
then 2**32 or 4294967296 possible outcomes.
But you asked for combinations, not permutations.
For this example, there can only be 33 possible
combinations:

those with 0 1's,
those with 2 1's,
those with 3 1's,
....,
those with 31 1's and
those with 32 1's.

So if 4 billion base 2 permutations reduces to
33 base 2 combinations, why do you think it
unreasonable that 10 billion base 10 permutations
reduces to 92378 base 10 combinations?

But just to be sure, I'm having my Permution/Combination
generator create all the combinations with
replacement. This takes quite a while, but it
finally came up with the answer:

92378

Would you like me to list them all?


Quote:

Stig Holmquist


A few extreme sets would be e.g. ten of the same digit or a mix of
only two digits or 0123456789.

Using a generator to find combinations w/replacement, I get for

2 balls - 3 combinations
3 balls - 10 combinations
4 balls - 35 combinations
5 balls - 126 combinations

and seeing where these fall on Pascal's triangle, the answer for
m balls is C(2m-1,m).


Suppose I calculate the std.dev. for each set and graph the
various frequencies. Would I get a chi-square curve with 9 df?

Uh...


Stig Holmquist
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Stig Holmquist
science forum beginner


Joined: 30 Apr 2005
Posts: 48

PostPosted: Tue Jul 11, 2006 11:29 pm    Post subject: Re: How many samlples from 10 digits? Reply with quote

On 11 Jul 2006 10:09:15 -0700, "mensanator@aol.com"
<mensanator@aol.com> wrote:

Quote:
Stig Holmquist wrote:
Suppose I draw one sample at a time with replacement from a bin with
10 balls numbered 0 to 9 and do this ten times. How many different
sets of ten can I generate disregarding order?

92378


A few extreme sets would be e.g. ten of the same digit or a mix of
only two digits or 0123456789.

Using a generator to find combinations w/replacement, I get for

2 balls - 3 combinations
3 balls - 10 combinations
4 balls - 35 combinations
5 balls - 126 combinations

and seeing where these fall on Pascal's triangle, the answer for
m balls is C(2m-1,m).


Suppose I calculate the std.dev. for each set and graph the
various frequencies. Would I get a chi-square curve with 9 df?

Uh...


Stig Holmquist

I reposted this problem at sci.math.num-analysis as:
Probability of 1234567890 and got entirely different
answer.

Stig Holmquist
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matt271829-news@yahoo.co.
science forum Guru


Joined: 11 Sep 2005
Posts: 846

PostPosted: Wed Jul 12, 2006 12:01 am    Post subject: Re: How many samlples from 10 digits? Reply with quote

Stig Holmquist wrote:
Quote:
On 11 Jul 2006 10:09:15 -0700, "mensanator@aol.com"
mensanator@aol.com> wrote:

Stig Holmquist wrote:
Suppose I draw one sample at a time with replacement from a bin with
10 balls numbered 0 to 9 and do this ten times. How many different
sets of ten can I generate disregarding order?

92378


A few extreme sets would be e.g. ten of the same digit or a mix of
only two digits or 0123456789.

Using a generator to find combinations w/replacement, I get for

2 balls - 3 combinations
3 balls - 10 combinations
4 balls - 35 combinations
5 balls - 126 combinations

and seeing where these fall on Pascal's triangle, the answer for
m balls is C(2m-1,m).


Suppose I calculate the std.dev. for each set and graph the
various frequencies. Would I get a chi-square curve with 9 df?

Uh...


Stig Holmquist

I reposted this problem at sci.math.num-analysis as:
Probability of 1234567890 and got entirely different
answer.


I agree with Mensanator's answer of 92378.
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Stephen J. Herschkorn
science forum Guru


Joined: 24 Mar 2005
Posts: 641

PostPosted: Wed Jul 12, 2006 3:11 am    Post subject: Re: How many samlples from 10 digits? Reply with quote

Stig Holmquist wrote:

Quote:
Suppose I draw one sample at a time with replacement from a bin with
10 balls numbered 0 to 9 and do this ten times. How many different
sets of ten can I generate disregarding order?

A few extreme sets would be e.g. ten of the same digit or a mix of
only two digits or 0123456789.

Suppose I calculate the std.dev. for each set and graph the
various frequencies. Would I get a chi-square curve with 9 df?



I am not sure exactly what your are describing. Suppose your draws are,
in order.

1 2 1 3 5 2 6 5 9 0

Is your observation the set

{0, 1, 2, 3, 5, 6, 9}

or the decuple

(0, 1, 1, 2, 2, 3, 5, 5, 6, 9)?


In the former case there are 2^10-1 = 1023 possible outcomes; this is
the number of nonempty subsets of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. In
the latter case, there are C(19,10) (i.e., "19 choose 10") = 92378
possbile outcomes; this is the number of ways of assigning nonnnegative
integers to ten variables so that their sum is ten. (Reference: Ross,
A First Course in Probability.)

In neither case is the distribution uniform. If you need to, specify
which you want, and we can help you with the actual distribution.

In your chi-squared test, you will have 1022 degrees of freedom in the
former case, 92377 in the latter, assuming you have a large enough
sample size to have one bin for each possible outcome. In either case,
the chi-squared distribution is close to normal. AIn practice, you will
likely want to collect some rare values in bins, in which case the
degrees of freedom are (is?) reduced.

--
Stephen J. Herschkorn sjherschko@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan
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Stig Holmquist
science forum beginner


Joined: 30 Apr 2005
Posts: 48

PostPosted: Wed Jul 12, 2006 11:44 pm    Post subject: Re: How many samlples from 10 digits? Reply with quote

On 11 Jul 2006 16:16:30 -0700, "mensanator@aol.com"
<mensanator@aol.com> wrote:

Quote:

Stig Holmquist wrote:
On 11 Jul 2006 10:09:15 -0700, "mensanator@aol.com"
mensanator@aol.com> wrote:

Stig Holmquist wrote:
Suppose I draw one sample at a time with replacement from a bin with
10 balls numbered 0 to 9 and do this ten times. How many different
sets of ten can I generate disregarding order?

92378

This number does not seem realistic considering that the cchances
of drawing the same digit 10 times is only 1 per 10^10

Why do you think that?

Suppose the urn contained only two balls 0 & 1
and you selected 32 with replacement. There are
then 2**32 or 4294967296 possible outcomes.
But you asked for combinations, not permutations.
For this example, there can only be 33 possible
combinations:

those with 0 1's,
those with 2 1's,
those with 3 1's,
...,
those with 31 1's and
those with 32 1's.

So if 4 billion base 2 permutations reduces to
33 base 2 combinations, why do you think it
unreasonable that 10 billion base 10 permutations
reduces to 92378 base 10 combinations?

But just to be sure, I'm having my Permution/Combination
generator create all the combinations with
replacement. This takes quite a while, but it
finally came up with the answer:

92378

Would you like me to list them all?

I'm grateful, impressed and intrigued by your 92378 data.

I would hate to see them go to waste and would like to ask
you to calculate the std.dev for each set and list them so
that a graph may be condtructed. The values would
range from 0 to 4.743 and some kind of curve would
be obtained. Maybe a chi-squared type. If so it
would have a mean amd a std.dev. of 2 times the mean.

I'm sure this work could be published in a statistcal
publication as a novel contrbution.

Hope you will do this and report the result here.

Stig Holmquist>
Quote:

Stig Holmquist


A few extreme sets would be e.g. ten of the same digit or a mix of
only two digits or 0123456789.

Using a generator to find combinations w/replacement, I get for

2 balls - 3 combinations
3 balls - 10 combinations
4 balls - 35 combinations
5 balls - 126 combinations

and seeing where these fall on Pascal's triangle, the answer for
m balls is C(2m-1,m).


Suppose I calculate the std.dev. for each set and graph the
various frequencies. Would I get a chi-square curve with 9 df?

Uh...


Stig Holmquist
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