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#1 2013-08-08 02:24:41

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Shoelace

A shoelace is lying on the floor, and attached you can see its shadow. If I pull it, what is the probability that it will produce a knot?

View Image: knot.jpg

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#2 2013-08-08 03:03:53

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,518

Re: Shoelace

Hi anna_gg


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#3 2013-08-08 03:18:25

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Re: Shoelace

All I can tell is that there are 7 intersections, so there are 2^7 =128 possible ways the shoelace can cross itself. Of these, we should keep only the ones that are "over/under/over" and "under/over/under", which will produce a knot. But I cannot figure out how many these are.

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#4 2013-08-08 03:30:46

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,518

Re: Shoelace

Um, have you seen my answer?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#5 2013-08-08 03:46:38

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Re: Shoelace

anonimnystefy wrote:

Um, have you seen my answer?

Yes, but I do not understand how you calculate it smile

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#6 2013-08-08 06:29:18

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,518

Re: Shoelace

Well, I looked at the two parts separately, because they are virtually independent. For one, there are 8 possibilities for one and 16 for the other. Then I just combine them.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#7 2013-08-09 05:47:55

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Re: Shoelace

anonimnystefy wrote:

Well, I looked at the two parts separately, because they are virtually independent. For one, there are 8 possibilities for one and 16 for the other. Then I just combine them.

So you mean that in the right part there are only 3/16 possibilities that a knot is produced? Because at the left part I see there are 2/8, that is 4/16.

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#8 2013-08-09 05:51:50

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,518

Re: Shoelace

No, both probabilities are 1/4.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#9 2013-08-09 06:48:28

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Re: Shoelace

anonimnystefy wrote:

No, both probabilities are 1/4.

So, how do you get 7/16? Am confused smile

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#10 2013-08-12 04:51:14

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Re: Shoelace

Got it. You don't add the two sections, you multiply them smile

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#11 2013-08-12 04:55:28

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,233

Re: Shoelace

How did you get 7 / 16 by multiplying?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#12 2013-08-12 19:28:11

anna_gg
Member
Registered: 2012-01-10
Posts: 113

Re: Shoelace

bobbym wrote:

How did you get 7 / 16 by multiplying?

The probability for both parts NOT to produce a knot is 3/4 for each (since the probability to produce a knot is 1/4). Therefore the total probability NOT to have a knot is 3/4*3/4 = 9/16, thus giving us probability to have a knot 1-9/16 = 7/16.

It wasn't me who solved it, I just unraveled anonimnystefy's solution smile

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#13 2013-08-12 20:09:32

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,233

Re: Shoelace

Hi;

Okay, thanks for the answer.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#14 2013-08-18 01:32:09

bohr
Member
Registered: 2013-08-18
Posts: 1

Re: Shoelace

I understand that the left part has a probability of 1/4 to produce a knot, since only two of the eight possible ways produce a knot (over/under/over and under/over/under). However, how the right part has also 1/4 probability; You mean that from the 16 possible ways, 4 of them produce a knot;

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