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#1 2013-04-13 01:00:46

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
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xpected value

A school is running a raffle for two prizes. 59 tickets were sold for the raffle, numbered 1,…,59. All the tickets are put into a hat and a teacher picks out two tickets which have numbers i and j from the hat. What is the expected value of |i−j|?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#2 2013-04-13 01:23:45

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 88,665

Re: xpected value

Hi;


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#3 2013-04-13 01:28:26

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,950
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Re: xpected value

I am getting 20


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#4 2013-04-13 01:31:22

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 88,665

Re: xpected value

There is no replacement?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#5 2013-04-13 01:40:55

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,601

Re: xpected value

I am getting 20 as well.


“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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#6 2013-04-13 01:43:49

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,665

Re: xpected value

If there picking without replacement then the answer is 20. If not the answer in post #2 holds.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#7 2013-04-13 01:46:10

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,601

Re: xpected value

"The teacher picks out two tickets"


“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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#8 2013-04-13 01:50:13

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,665

Re: xpected value

That does suggest without replacement. The answer is (59 + 1 ) / 3 = 20.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#9 2013-04-13 01:52:17

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,950
Website

Re: xpected value

Explain in simple words please hmm


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#10 2013-04-13 01:56:48

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 88,665

Re: xpected value

Hi;

This is the answer from my notes.

There are n(n-1) ways to pick two numbers from the unit interval 1 ... n.

If they are ordered high and low then there are:

ways.

The standard way now to do this is to look at the ordered pairs. There is 1 way for 2 to be the maximum and 2 ways for 3 to be the maximum and there are generally m-1 ways for m to be the maximum. Using the formula for expected value or expected number:

For the second part, you do it in the same way. There are
still n (n-1)/2 ways to arrange n numbers with high and low.

Now you go through the numbers in the same way. 1 is the lowest n-1 times. 2 is the lowest n - 2 times. n can never be the lowest. n-1 is the lowest one time so generally m is lowest n - m times. We use the same formula for expected value:

Now for the difference.

Just subtract the expected value of the lowest from the highest.

The absolute value is implied in there.


When you have time please sign up over here:

https://brilliant.org/


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#11 2013-04-15 01:31:27

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: xpected value

Explain in simple words please hmm

Hi,

Let's begin with the list of |i - j| possibilities :
1, 2, 3, 4, ... , 58
1, 2, 3, 4 ..  57
1, 2, 3, .. 56
.
.
.
1, 2
1

Each number from every row is equiprobable
Hence, the expectation:

and for any n


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#12 2013-04-15 01:46:16

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,950
Website

Re: xpected value

Hmm, and the rows are obtained how?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#13 2013-04-15 01:53:04

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: xpected value

Start with the first two numbers and end with the last two, listing each combination.

(2-1), (3-1), ... (59-1)
(3-2), (4-2) ... (59-2)
.
.
(58-57), (59-57)
(59-58)


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#14 2013-04-15 02:07:50

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 17,950
Website

Re: xpected value

gOt it
Thanks smile

Last edited by Agnishom (2013-04-15 02:08:24)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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