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## #1 2013-01-31 12:40:55

cooljackiec
Member
Registered: 2012-12-13
Posts: 185

### Please Post By 10:30 Eastern Time!!!

A bag has marbles of 4 colors: red, white, blue, and green. Assume that if we take four marbles out at random (without replacement), each of the following is equally likely:

(1) one marble of each color is chosen,

(2) one white, one blue, and two reds are chosen,

(3) one blue and three reds are chosen,

(4) all four are red.

What is the smallest possible number of marbles in the bag?

I see you have graph paper.
You must be plotting something

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## #2 2013-01-31 20:56:58

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 103,751

### Re: Please Post By 10:30 Eastern Time!!!

Hi;

Sorry, I could not meet the time, I was sleeping.

The smallest I can come up with is a bag containing 11 reds, 3 whites, 2 blues and 5 greens.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
A number by itself is useful, but it is far more useful to know how accurate or certain that number is.

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## #3 2013-02-01 11:48:49

cooljackiec
Member
Registered: 2012-12-13
Posts: 185

### Re: Please Post By 10:30 Eastern Time!!!

Its okay, I got 21

I see you have graph paper.
You must be plotting something

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## #4 2013-02-01 18:45:50

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 103,751

### Re: Please Post By 10:30 Eastern Time!!!

Sorry, that is what I got too, I meant 3 whites not 4!

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
A number by itself is useful, but it is far more useful to know how accurate or certain that number is.

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## #5 2014-02-06 14:11:48

Tpetrie01
Guest

### Re: Please Post By 10:30 Eastern Time!!!

Sorry, I know this post is old - but can you give me a hint as to how you figured this problem out, bobbym? I've got a similar problem and i want to know which direction i should be thinking in.

## #6 2014-02-06 14:20:35

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 103,751

### Re: Please Post By 10:30 Eastern Time!!!

Hi;

You would use the multivariate hypergeometric distribution. After that you would try it until you found the right number. This would require some trial and error best done by computer.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
A number by itself is useful, but it is far more useful to know how accurate or certain that number is.

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## #7 2014-02-06 14:39:40

Tpetrie01
Guest

### Re: Please Post By 10:30 Eastern Time!!!

Thanks! This helps a lot.

## #8 2014-02-06 14:42:14

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 103,751

### Re: Please Post By 10:30 Eastern Time!!!

You know how to use the multivariate hypergeometric?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
A number by itself is useful, but it is far more useful to know how accurate or certain that number is.

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## #9 2014-12-21 10:22:26

deoxysxxxx
Member
Registered: 2014-12-21
Posts: 9

### Re: Please Post By 10:30 Eastern Time!!!

Is there an easy way to solve it without using some complex formula thing?

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## #10 2014-12-21 12:04:26

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 103,751

### Re: Please Post By 10:30 Eastern Time!!!

Hi;

I do not see how you can avoid either more complicated math or some trial and error. This is a fairly tough problem to me. Where is it being asked?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
A number by itself is useful, but it is far more useful to know how accurate or certain that number is.

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## #11 2014-12-21 12:08:22

deoxysxxxx
Member
Registered: 2014-12-21
Posts: 9

### Re: Please Post By 10:30 Eastern Time!!!

An online class called Art Of Problem Solving.  Have you heard of it?

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## #12 2014-12-21 12:14:41

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 103,751

### Re: Please Post By 10:30 Eastern Time!!!

I am a member of the AOPS. I have never seen any solution to this posted over there.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
A number by itself is useful, but it is far more useful to know how accurate or certain that number is.

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## #13 2014-12-22 03:46:19

deoxysxxxx
Member
Registered: 2014-12-21
Posts: 9

### Re: Please Post By 10:30 Eastern Time!!!

That's weird. It's problem number 5, week 12 for Introduction to Counting and Probability

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## #14 2014-12-22 07:54:06

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 103,751

### Re: Please Post By 10:30 Eastern Time!!!

What solution do they post for it?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
A number by itself is useful, but it is far more useful to know how accurate or certain that number is.

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## #15 2014-12-22 10:02:58

deoxysxxxx
Member
Registered: 2014-12-21
Posts: 9

### Re: Please Post By 10:30 Eastern Time!!!

I need to wait for it to be posted xD

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## #16 2014-12-22 11:43:52

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,374

### Re: Please Post By 10:30 Eastern Time!!!

Hi;

The problem appears here (#15.23), with the solution below it.

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #17 2014-12-22 12:09:15

deoxysxxxx
Member
Registered: 2014-12-21
Posts: 9

### Re: Please Post By 10:30 Eastern Time!!!

I actually solved it this morning. Wow, my solution was the same as that one!

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## #18 2014-12-22 12:56:27

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 103,751

### Re: Please Post By 10:30 Eastern Time!!!

It is much easier by computer.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
A number by itself is useful, but it is far more useful to know how accurate or certain that number is.

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## #19 2014-12-22 21:37:28

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,374

### Re: Please Post By 10:30 Eastern Time!!!

Hi Bobby,

I ran 200,000-iteration simulation in Excel approx 5 times for each of the A, B and C scenarios, with the following results:

``````                                       rwbg  |   wbrr  |   brrr  |  rrrr
-----------------------------------
A. For r = 11, w = 3, b = 2, g = 5:    19%   |   21%   |   25%   |  35%
B. For r = 10, w = 3, b = 2, g = 5:    22%   |   22%   |   25%   |  31%
C. For r =  9, w = 3, b = 2, g = 5:    26%   |   24%   |   24%   |  26%``````

If I'm reading that right, C gives the most even spread across the four marble groups, and if so, then my results differ from everyone else's. But I could easily have muffed something because I haven't learnt probabilities...which is one reason for using E instead of M.

The percentage figures relate to the number of times each marble group was successfully chosen in the random selections.

Last edited by phrontister (2014-12-22 21:48:04)

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #20 2014-12-22 21:45:40

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 103,751

### Re: Please Post By 10:30 Eastern Time!!!

Hi;

If I'm reading that right, C gives the best result,

Why?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
A number by itself is useful, but it is far more useful to know how accurate or certain that number is.

Offline

## #21 2014-12-22 21:51:13

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,374

### Re: Please Post By 10:30 Eastern Time!!!

...each of the following is equally likely

I thought that meant that the four groups should all be chosen approx the same number of times in a simulation.

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #22 2014-12-22 22:01:09

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 103,751

### Re: Please Post By 10:30 Eastern Time!!!

Hi;

It means that if you had the some composition of balls in that urn and you picked 4, each of these choices

(1) one marble of each color is chosen,

(2) one white, one blue, and two reds are chosen,

(3) one blue and three reds are chosen,

(4) all four are red.

would be equally likely.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
A number by itself is useful, but it is far more useful to know how accurate or certain that number is.

Offline

## #23 2014-12-22 22:06:17

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,374

### Re: Please Post By 10:30 Eastern Time!!!

Yes, that's what I understood it to mean, but I thought that a simulation would work on this problem.

I tried various colour combinations and numbers of marbles, but the A, B and C scenarios gave the best results.

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #24 2014-12-22 22:14:27

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 103,751

### Re: Please Post By 10:30 Eastern Time!!!

Let's take a look at A:

For r = 11, w = 3, b = 2, g = 5:

Your chart shows that rwbg, wbrr, brrr,  rrrr all have different percentages. You must find the number of r's, w's, b's and g's that when you draw rwbg, wbrr, brrr,  rrrr  from that urn the percentages are the same.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
A number by itself is useful, but it is far more useful to know how accurate or certain that number is.

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## #25 2014-12-22 22:25:33

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,374

### Re: Please Post By 10:30 Eastern Time!!!

My simulation method could never give the same answers for the four groups.

Maybe my 5 times simulation @ 200000 iterations isn't enough. I keep getting similar results for each simulation group, so my code might be out.

Is an accurate simulation possible?

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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