Probability: Placing books into piles

In summary: There are 5 possibilities in terms of being distinguishable:books are piles arebooks are piles aren'tbooks aren't piles arebooks aren't piles aren't
  • #1
TranscendArcu
285
0

Homework Statement



Ten books are made into two piles. In how many ways can this be done if books as well as piles may or may not be distinguishable?

The Attempt at a Solution



So I thought I'd try what I thought was the easiest case first, that being that neither the books nor the piles are distinguishable. In that case, can't the books only be arranged one way? I wouldn't be able to tell the difference between the books in the pile nor the piles themselves, so all combinations would be identical. Is that right?
 
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  • #2
I don't think so. You can still distinguish between the case where there is 1 book in one pile and 9 books in the other pile from the case where there are two books in one pile and 8 books in the other pile.
 
  • #3
TranscendArcu said:

Homework Statement



Ten books are made into two piles. In how many ways can this be done if books as well as piles may or may not be distinguishable?

The Attempt at a Solution



So I thought I'd try what I thought was the easiest case first, that being that neither the books nor the piles are distinguishable. In that case, can't the books only be arranged one way? I wouldn't be able to tell the difference between the books in the pile nor the piles themselves, so all combinations would be identical. Is that right?

No. In that case, you still know there is a certain number of books in some pile. You just don't know if it is pile 1 or pile 2.

So if you can distinguish the piles, you can have 6 books in pile 1 or 6 books in pile 2. It cuts the possibilities about in half from the case where you cannot distinguish books but can distinguish piles. Since this one has so few cases, you can just count them:
9 1
8 2
7 3
6 4
5 5So there are 5 possibilities. Note, if you can tell the difference in piles, you would also need to include
4 6
3 7
2 8
1 9

There is also an area of ambiguity in the question. If you put 0 books in a pile, is that still a pile? I'm not sure whether the question wants you to include 10 0.
 
  • #4
RoshanBBQ said:
No. In that case, you still know there is a certain number of books in some pile. You just don't know if it is pile 1 or pile 2.

So if you can distinguish the piles, you can have 6 books in pile 1 or 6 books in pile 2. It cuts the possibilities about in half from the case where you cannot distinguish books but can distinguish piles. Since this one has so few cases, you can just count them:
9 1
8 2
7 3
6 4
5 5


So there are 5 possibilities. Note, if you can tell the difference in piles, you would also need to include
4 6
3 7
2 8
1 9

There is also an area of ambiguity in the question. If you put 0 books in a pile, is that still a pile? I'm not sure whether the question wants you to include 10 0.

Even though you cannot distinguish between the piles, there is still the issue as to how to count the piles. In "classical" probability there are 9 ways of doing it, whether or not you can distinguish between the piles (or rather, whether or not you bother to distinguish between them). In the quantum world there would only be 5 ways, because you could not distinguish between the piles *even if you wanted to*. So, in this problem, do we have not distinguished (but distinguishable-in-principle) piles, or do we have truly absolutely indistinguishable piles? The counts are different, and that makes the probabilities different.

RGV
 
  • #5
Ray Vickson said:
Even though you cannot distinguish between the piles, there is still the issue as to how to count the piles. In "classical" probability there are 9 ways of doing it, whether or not you can distinguish between the piles (or rather, whether or not you bother to distinguish between them). In the quantum world there would only be 5 ways, because you could not distinguish between the piles *even if you wanted to*. So, in this problem, do we have not distinguished (but distinguishable-in-principle) piles, or do we have truly absolutely indistinguishable piles? The counts are different, and that makes the probabilities different.

RGV

There are 4 possibilities in terms of being distinguishable:
books are piles are
books are piles aren't
books aren't piles are
books aren't piles aren't

So if in classical probability, there are 9 possibilities when books and piles are not distinguishable, how many possibilities are there when books are not distinguishable but piles are distinguishable?
 

1. What is the definition of probability?

The concept of probability is a measure of the likelihood of an event occurring. It is represented as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. In simpler terms, probability is the chance of something happening.

2. How is probability used in placing books into piles?

Probability is used to determine the likelihood of a book being placed in a specific pile. For example, if there are 10 books and 2 piles, each book has a 50% probability of being placed in either pile. This can be calculated by dividing the number of desired outcomes (placing the book in a specific pile) by the total number of possible outcomes (the number of piles).

3. Can probability be used to predict the exact placement of books?

No, probability cannot accurately predict the exact placement of books. It can only give an estimate of the likelihood of a book being placed in a certain pile. Other factors such as randomness and human error can affect the placement of books.

4. How does the number of books and piles affect the probability of placement?

The more books and piles there are, the lower the probability of a book being placed in a specific pile. This is because the number of possible outcomes increases, making it less likely for a book to be placed in a specific pile. For example, if there are 10 books and 10 piles, each book has a 10% chance of being placed in a specific pile.

5. How is probability related to statistics?

Probability is a fundamental concept in statistics. It is used to analyze and interpret data, make predictions, and draw conclusions about a population based on a sample. In the case of placing books into piles, probability can help us understand the likelihood of certain books being placed in specific piles based on a sample of books.

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