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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;

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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'

'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 97,106

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
- Registered: 2009-04-12
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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.**

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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: 97,106

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.**

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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'

'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 97,106

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?

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

'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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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'

'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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