Square dartboard puzzle
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4 members have voted
March 8th, 2020 at 10:39:46 AM
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There is a square dartboard of dimensions 1 by 1. A dart is thrown at it such that it can land anywhere with equal probability. Let the coordinates of where is lands be (x,y), where both x and y are uniformly and independently distributed from 0 to 1.
Let z = round(x/y)*. In other words, z = x/y, rounded to the nearest integer.
What is the probability that z is even?
Usual rules:
Notes:
* corrected. Previous formula incorrect.
Let z = round(x/y)*. In other words, z = x/y, rounded to the nearest integer.
What is the probability that z is even?
Usual rules:
- Please don't just plop a URL to a solution elsewhere until a winner here has been declared.
- All those who have won a beer previously are asked to not post answers or solutions for 24 after this posting. Past winners who must chime in early, may PM me.
- Beer to the first satisfactory answer and solution, subject to rule 2.
- Please put answers and solutions in spoiler tags.
Notes:
* corrected. Previous formula incorrect.
Last edited by: Wizard on Mar 8, 2020
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 8th, 2020 at 3:07:42 PM
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Looks trivial with a computer, and difficult but possible by hand. I'll shut up until tomorrow because I already won.
March 8th, 2020 at 3:11:00 PM
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I'll have a go, but I suspect it involves calculus (integration) .I'm rubbish at that, though.Quote: Wizard
What is the probability that z is even?
Psalm 25:16
Turn to me and be gracious to me, for I am lonely and afflicted.
Proverbs 18:2
A fool finds no satisfaction in trying to understand, for he would rather express his own opinion.
March 8th, 2020 at 3:45:37 PM
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Quote: OnceDearI'll have a go, but I suspect it involves calculus (integration) .I'm rubbish at that, though.
I solved it without calculus.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 8th, 2020 at 4:49:04 PM
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Quote: WizardThere is a square dartboard of dimensions 1 by 1. A dart is thrown at it such that it can land anywhere with equal probability. Let the coordinates of where is lands be (x,y), where both x and y are uniformly and independently distributed from 0 to 1.
Let z = (int)(x/y). In other words, z = x/y, rounded to the nearest integer.
What is the probability that z is even?
I object to misdirection tactic: The Int() function does not round to the nearest integer: It always rounds down!
Function Round(,0) rounds nearest!
I'm going to progressively break down the target into squares and hit the centre of each square with a dart.
1 x 1 array of squares hit in the middle.
x=y therefore z=1 = odd with 0% probability of even.
2 x 2 array each hit in the middle
x1=0.25, y1=0.25, z=1 is odd
x2=0.25, y2=0.75, z=0 is even
x3=0.75, y3=0.25, z=3 is odd
x4=0.75, y4=0.75, z=1 is odd
z is even 1/4 of the time therefore 25% probability
3x3 array
x1=1/6, y1=1/6, z=1 is odd
x2=1/6, y2=3/6, z=0 is even
x3=1/6, y3=5/6, z=0 is even
x4=3/6, y1=1/6, z=3 is odd
x5=3/6, y2=3/6, z=1 is odd
x6=3/6, y3=5/6, z=0 is even
x7=5/6, y1=1/6, z=5 is odd
x8=5/6, y2=3/6, z=1 is odd
x9=5/6, y3=5/6, z=1 is odd
z is even 3/9 of the time therefore 33.3% probability
4 x 4 array
Trust me and Excel
z is even 6/16 of the time therefore 37.5% probability
HMMM Are we converging on 50%?
10 x 10 array
Trust me and Excel
z is even 43/100 of the time therefore 43% probability
HMMM Are we converging on 50%?
100 x 100 array
Trust me and Excel
z is even 4606/10000 of the time therefore 46.06% probability
HMMM Are we converging on 50%?
1000 x 1000 array
Trust me and Excel
z is even 464220/1000000 of the time therefore 46.42% probability
HMMM Are we converging SLOWLY on 50%
I'm going to guess 50% will be the eventual answer.
Damned if I can do integration though
1 x 1 array of squares hit in the middle.
x=y therefore z=1 = odd with 0% probability of even.
2 x 2 array each hit in the middle
x1=0.25, y1=0.25, z=1 is odd
x2=0.25, y2=0.75, z=0 is even
x3=0.75, y3=0.25, z=3 is odd
x4=0.75, y4=0.75, z=1 is odd
z is even 1/4 of the time therefore 25% probability
3x3 array
x1=1/6, y1=1/6, z=1 is odd
x2=1/6, y2=3/6, z=0 is even
x3=1/6, y3=5/6, z=0 is even
x4=3/6, y1=1/6, z=3 is odd
x5=3/6, y2=3/6, z=1 is odd
x6=3/6, y3=5/6, z=0 is even
x7=5/6, y1=1/6, z=5 is odd
x8=5/6, y2=3/6, z=1 is odd
x9=5/6, y3=5/6, z=1 is odd
z is even 3/9 of the time therefore 33.3% probability
4 x 4 array
Trust me and Excel
z is even 6/16 of the time therefore 37.5% probability
HMMM Are we converging on 50%?
10 x 10 array
Trust me and Excel
z is even 43/100 of the time therefore 43% probability
HMMM Are we converging on 50%?
100 x 100 array
Trust me and Excel
z is even 4606/10000 of the time therefore 46.06% probability
HMMM Are we converging on 50%?
1000 x 1000 array
Trust me and Excel
z is even 464220/1000000 of the time therefore 46.42% probability
HMMM Are we converging SLOWLY on 50%
I'm going to guess 50% will be the eventual answer.
Damned if I can do integration though
Last edited by: OnceDear on Mar 8, 2020
Psalm 25:16
Turn to me and be gracious to me, for I am lonely and afflicted.
Proverbs 18:2
A fool finds no satisfaction in trying to understand, for he would rather express his own opinion.
March 8th, 2020 at 6:30:56 PM
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P(z=even) = 0.4646 ?
March 8th, 2020 at 6:37:36 PM
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Quote: OnceDearI object to misdirection tactic: The Int() function does not round to the nearest integer: It always rounds down!
You're absolutely right. I meant z=round(x/y).
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 8th, 2020 at 6:38:42 PM
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Quote: ssho88P(z=even) = 0.4646 ?
I agree with this answer to four digits. However, to get full credit, and the beer, I need to see an expression of the answer as well as a solution. In other words, show your work.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 8th, 2020 at 9:02:53 PM
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Can't show the step by step solution, just pure simulation results. LOL
March 8th, 2020 at 9:04:08 PM
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Fun problem.
I get (5-pi)/4
Not trying to win a beer, just having fun.
I heart Crystal Math.
March 8th, 2020 at 9:24:56 PM
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Quote: CrystalMath
Fun problem.
I get (5-pi)/4
Not trying to win a beer, just having fun.
Long time no see! I agree with your answer.
To win the beer, one must show their work. Lacking a solution, all except past beer winners are still eligible to win.
To win the beer, one must show their work. Lacking a solution, all except past beer winners are still eligible to win.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 9th, 2020 at 7:26:20 AM
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In a bit over three hours the 24-hour delay for past winners will be lifted. Remember, to get full credit and the beer, you must show your work.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 9th, 2020 at 11:42:20 AM
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If the dart hits the board, the probability of hitting a region is the area of that region divided by the total area of the board, which is (1)(1) = 1.
The even values of z are: 0, 2, 4, 6, etc.
The part of the board with z equal to 0 is the region bounded by the lines with equations x=0, y=1, and x=y/2. The area of this region is (1/2)(1/2) = 1/4
The part of the board with z equal to 2 is the region bounded by the lines with equations of x = 3y/2, x =5y/2, and x=1. Its area is half its base times its height, which is (1/2)(2/3-2/5)=1/3 – 1/5.
Likewise, the area with z = 4, is the region between x=7y/2, x=9y/2, and x=1. Its area is 1/7 – 1/9.
The area of the region z=6 is 1/11 – 1/13.
You can see the pattern. Call the sum of the areas P. Then P = ¼ + (1/3 – 1/5 + 1/7 – 1/9 + 1/11 – 1/13 + 1/15 – 1/17+…).
The above expression can be approximated with a calculator or Excel, but we can use a famous series:
pi/4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + 1/13 – 1/15 + 1/17 - …
to find an exact expression for the probability:
P = ¼ + (1/3 – 1/5 + 1/7 – 1/9 + 1/11 – 1/13 + 1/15 – 1/17+…) = ¼ + (1-pi/4) = (5-pi)/4, which is about 0.464602.
The even values of z are: 0, 2, 4, 6, etc.
The part of the board with z equal to 0 is the region bounded by the lines with equations x=0, y=1, and x=y/2. The area of this region is (1/2)(1/2) = 1/4
The part of the board with z equal to 2 is the region bounded by the lines with equations of x = 3y/2, x =5y/2, and x=1. Its area is half its base times its height, which is (1/2)(2/3-2/5)=1/3 – 1/5.
Likewise, the area with z = 4, is the region between x=7y/2, x=9y/2, and x=1. Its area is 1/7 – 1/9.
The area of the region z=6 is 1/11 – 1/13.
You can see the pattern. Call the sum of the areas P. Then P = ¼ + (1/3 – 1/5 + 1/7 – 1/9 + 1/11 – 1/13 + 1/15 – 1/17+…).
The above expression can be approximated with a calculator or Excel, but we can use a famous series:
pi/4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + 1/13 – 1/15 + 1/17 - …
to find an exact expression for the probability:
P = ¼ + (1/3 – 1/5 + 1/7 – 1/9 + 1/11 – 1/13 + 1/15 – 1/17+…) = ¼ + (1-pi/4) = (5-pi)/4, which is about 0.464602.
March 9th, 2020 at 2:04:21 PM
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Just saw this thread. I believe (unofficially) that Chester beat us all to the answer.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
March 9th, 2020 at 5:40:06 PM
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Quote: gordonm888I believe (unofficially) that Chester beat us all to the answer.
Yes, let's make it official. CD, I owe you another beer. I lost count how many we're at.
Yes, the key to this one is knowing Leibniz formula for π.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 11th, 2020 at 12:28:24 AM
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Indeeed, as CrystalMath showed, the probability of getting an even integer is (5-pi)/4 if x/y is rounded to the nearest integer. This probability becomes 1- ln(2)/2 if x/y is rounded down to the nearest integer, as can be calculated by using integral calculus and the series expansion ln(2)=1-1/2+1/3-1/4+. . . ; otherwise, you may check the answer by Monte Carlo simulation.
Last edited by: Klopp on Mar 11, 2020
