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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|?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Hi;

**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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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'

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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There is no replacement?

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
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I am getting 20 as well.

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**bobbym****Administrator**- From: Bumpkinland
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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.****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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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
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"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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,768

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.****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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Explain in simple words please

'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'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,768

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.

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

Explain in simple words please

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

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Hmm, and the rows are obtained how?

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

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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gOt it

Thanks

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

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

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