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**samuel.bradley.99****Member**- Registered: 2016-03-16
- Posts: 51

In a stack of 31 identical books, I am searching 5 which contain a dedication by an old friend, written in the first page. How many books do I have to check, on average, in order to locate the ones I am looking for?

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

Are you searching for 5 books that have this dedication or are you searching 5 books at a time?

**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.**

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**samuel.bradley.99****Member**- Registered: 2016-03-16
- Posts: 51

I am searching for these particular 5 books, of the stack of 31. This means I start picking books at random, open the cover to see if they have the dedication and then put them aside, to continue with the remaining. I can be very lucky and find all 5 at my first few attempts, or have to check all 31 books until I get them all.

We are asking for the average number of books that I must check.

bobbym wrote:

Are you searching for 5 books that have this dedication or are you searching 5 books at a time?

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

Are there only 5 books in the bunch or are there more.

**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.**

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**samuel.bradley.99****Member**- Registered: 2016-03-16
- Posts: 51

I don't know what you mean by "bunch"...

I pick one book at a time, check for the dedication and then continue. As soon as I find all 5, I stop.

bobbym wrote:

Are there only 5 books in the bunch or are there more.

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

All 5, that implies there are only 5 of these books that have the dedication. That is correct?

**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.**

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**samuel.bradley.99****Member**- Registered: 2016-03-16
- Posts: 51

Ahh yes. Only 5 have the dedication.

bobbym wrote:

All 5, that implies there are only 5 of these books that have the dedication. That is correct?

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

Hi;

I am getting an expected value of 80 / 3 checks to find all 5.

**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.**

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**samuel.bradley.99****Member**- Registered: 2016-03-16
- Posts: 51

Dear Bobby,

Only by intuition, I would expect a smaller number, although I don't know the answer.

Can you share solution?

bobbym wrote:

Hi;

I am getting an expected value of 80 / 3 checks to find all 5.

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

Intuition has no place in expectation but even so the expected amount for just one book out of 31 is around 16 so 80 / 3 for 5 is not unbelievable.

The answer was derived in 3 ways:

1) A simulation produced 26.666151 as well as a possible formula.

2) I then checked every possible arrangement of books there are only

arrangements and got an answer of 80 / 3.

3) There is a distribution for problems like this, it is called the negative hypergeometric distribution. Using it I get

**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.**

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**samuel.bradley.99****Member**- Registered: 2016-03-16
- Posts: 51

Dear Bobby,

I agree that intuition is not the proper way of solving math problems

I trust your solution is correct. Can you please explain No 2 an in particular, how you got 80/3 from the total number of possible arrangements? If I understand well, this is the number of ways to select 5 out of 31.

Many thanks,

Sam

bobbym wrote:

Intuition has no place in expectation but even so the expected amount for just one book out of 31 is around 16 so 80 / 3 for 5 is not unbelievable.

The answer was derived in 3 ways:

1) A simulation produced 26.666151 as well as a possible formula.

2) I then checked every possible arrangement of books there are only

arrangements and got an answer of 80 / 3.

3) There is a distribution for problems like this, it is called the negative hypergeometric distribution. Using it I get

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

I agree that intuition is not the proper way of solving math problems

I do not. Without our intuition we would not be able to solve any problem at all.

I used a computer for both 1 and 2. For 2

I generated all possible collections of 31 books and then just calculated the exact answer from that.

**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.**

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