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Old 14.11.2011, 15:16
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A question for the math geeks

Take the interval of real numbers [0,100].

From this interval, n different numbers are picked randomly, where n>1.

What is the expected difference between the largest and smallest number picked for a given value of n? (assume uniform distribution, where required)

Happy to hear an answer for the case n=2, as well as the more general case for arbitrary integer values of n>1.

Last edited by Phil_MCR; 14.11.2011 at 15:38.
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Old 14.11.2011, 15:17
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Re: A question for the math geeks

I'll take a SWAG and say 100...
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Old 14.11.2011, 15:25
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Re: A question for the math geeks

98 ?????


cheers
SC
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Old 14.11.2011, 15:32
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Re: A question for the math geeks

would it be 100-n-2?
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Old 14.11.2011, 15:50
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Re: A question for the math geeks

well, for n=2, by 'common sense' you might guess the answer to be around 50.

i tried to use a hand-waving geometric argument and get an answer 37.5. which seems to be a reasonable answer and goes with my 'gut feeling'.

if it is right, i would be happy to see a rigorous proof and also to extend a result for n>2.

i'm sure the IT geeks can come up with an empirical answer by statistical trials.
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Old 14.11.2011, 15:53
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Re: A question for the math geeks

example for n=2. pick the first number at random, it is 35. pick the 2nd number at random it is 67. difference between the two is: 32.

to get an answer of 98, you'd have to pick: 0, 98; 1, 99, or 2, 100 (just taking whole number examples). so 98 doesn't seem like a likely answer
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Old 14.11.2011, 15:55
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Re: A question for the math geeks

100 * (n-1) / (n+1) has my money.
and Excel seems to agree.

oops. edited again... typing while working spreadhseet :-)
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Old 14.11.2011, 15:58
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Re: A question for the math geeks

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100 *n / (n+1) has my money.
and Excel seems to agree.
Nice ninja edit
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Old 14.11.2011, 16:40
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Re: A question for the math geeks

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Take the interval of real numbers [0,100].

From this interval, n different numbers are picked randomly, where n>1.

What is the expected difference between the largest and smallest number picked for a given value of n? (assume uniform distribution, where required)

Happy to hear an answer for the case n=2, as well as the more general case for arbitrary integer values of n>1.
The expected range is 100*(N-1)/(N+1)

The probability density function of the range r is a beta function, N*(N-1)*(1-r/100)*(r/100)^(n-2), so you can multiply that by r and integrate between 0 and 100 to get the expected range.

To test the answer, we can look at special cases:

N = 1, expected range is 0 (obviously correct)

As N goes to infinity, chances are the minimum is very close to 0 and the maximum is very close to 100, so the range will tend to 100.
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Old 14.11.2011, 16:50
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Re: A question for the math geeks

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The expected range is 100*(N-1)/(N+1)

The probability density function of the range r is a beta function, N*(N-1)*(1-r/100)*(r/100)^(n-2), so you can multiply that by r and integrate between 0 and 100 to get the expected range.

To test the answer, we can look at special cases:

N = 1, expected range is 0 (obviously correct)

As N goes to infinity, chances are the minimum is very close to 0 and the maximum is very close to 100, so the range will tend to 100.
thanks. this seems to make sense.
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Old 14.11.2011, 16:41
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There is no spoon. ;-)
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Old 15.11.2011, 16:00
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Re: A question for the math geeks

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There is no spoon. ;-)
Maybe a soup spoon ?
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Old 15.11.2011, 18:03
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Re: A question for the math geeks

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Maybe a soup spoon ?
In Switzerland? Not if this is to be believed.
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Old 14.11.2011, 23:37
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Re: A question for the math geeks

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But it's really what I said...in less words...and without an explanation
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100*n/(n+1) is what pops into my mind
nope: your answer was wrong.
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Old 14.11.2011, 23:52
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Re: A question for the math geeks

OK. Here's another question.

On a flat table, there is drawn a circle with radius r. A line is drawn through the circle running exactly north/south and passes directly through the centre of the circle.

A thin needle of length l, where 0<l<=r is thrown randomly onto the circle. if any part of the needle is outside the circle, it is picked up and re-thrown until it lies entirely on or within the circle.

What is the probability that the needle touches the line?

Second part of the question:

n>0 identical needles of lenght l are thrown in the same manner onto the circle. a heavy cylinder then is pressed onto the needles so that they are pressed into a flat plane.

let T be the number of different needles that touch or cross. for example, if 3 needles are thrown and needles 1 and 2 land in exactly the same way and the third lies across both 1 and 2, then T=3: 1 touches 2, 1 touches 3, 2 touches 3.

what is the expected value of T for a given value of n, l and r?
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Last edited by Phil_MCR; 18.11.2011 at 11:22.
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Old 15.11.2011, 00:01
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Re: A question for the math geeks

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Old 15.11.2011, 00:09
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Re: A question for the math geeks

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OK. Here's another question.
No. Do your own homework now!
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Old 15.11.2011, 00:11
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Re: A question for the math geeks

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OK. Here's another question.

On a flat table, there is drawn a circle with radius r. A line is drawn through the circle running exactly north/south and passes directly through the centre of the circle.

A thin needle of length l, where 0<l<=r is thrown randomly but in such a way that it always falls such that no part of it extends outside the circumference of the circle.

What is the probability that the needle touches the line?

Second part of the question:

n>0 identical needles are thrown in the same manner onto the circle. a heavy cylinder then is pressed onto the needles so that they are pressed into a flat plane.

let T be the number of different needles that touch or cross. for example, if 3 needles are thrown and needles 1 and 2 land in exactly the same way and the third lies across both 1 and 2, then T=3: 1 touches 2, 1 touches 3, 2 touches 3.

what is the expected value of T for a given value of n, l and r?
Just last week, I was confronted with exactly this dilemma while sitting at my desk, trying to improve my company's profitability. Few people understand the impact on corporate revenues of needles crossing from one half of a circle to the other, and even fewer, the devastating consequences of underestimating the number of needles that touch or cross when many needles are thrown with gay abandon into a circle just before the annual heavy-cylinder squashing ritual takes place.

I hope someone can provide the answer to this problem that has plagued corporate beancounters for centuries, but I only wish that answer had been available when I really needed it, last week.
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Old 15.11.2011, 13:18
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Re: A question for the math geeks

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Just last week, I was confronted with exactly this dilemma while sitting at my desk, trying to improve my company's profitability. Few people understand the impact on corporate revenues of needles crossing from one half of a circle to the other, and even fewer, the devastating consequences of underestimating the number of needles that touch or cross when many needles are thrown with gay abandon into a circle just before the annual heavy-cylinder squashing ritual takes place.

I hope someone can provide the answer to this problem that has plagued corporate beancounters for centuries, but I only wish that answer had been available when I really needed it, last week.
would have repped if i didn't run out of rep points to give
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Old 15.11.2011, 13:27
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Re: A question for the math geeks

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would have repped if i didn't run out of rep points to give
I can wait ...
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