Poll

2 votes (50%)
No votes (0%)
2 votes (50%)
No votes (0%)
2 votes (50%)
No votes (0%)
2 votes (50%)
1 vote (25%)
1 vote (25%)
No votes (0%)

4 members have voted

Wizard
Administrator
Wizard
  • Threads: 1491
  • Posts: 26435
Joined: Oct 14, 2009
March 8th, 2020 at 10:39:46 AM permalink
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:

  1. Please don't just plop a URL to a solution elsewhere until a winner here has been declared.
  2. 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.
  3. Beer to the first satisfactory answer and solution, subject to rule 2.
  4. 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)
scrooge
scrooge
  • Threads: 4
  • Posts: 39
Joined: Nov 22, 2016
March 8th, 2020 at 3:07:42 PM permalink
Looks trivial with a computer, and difficult but possible by hand. I'll shut up until tomorrow because I already won.
OnceDear
OnceDear
  • Threads: 63
  • Posts: 7471
Joined: Jun 1, 2014
March 8th, 2020 at 3:11:00 PM permalink
Quote: Wizard


What is the probability that z is even?

I'll have a go, but I suspect it involves calculus (integration) .I'm rubbish at that, though.
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.
Wizard
Administrator
Wizard
  • Threads: 1491
  • Posts: 26435
Joined: Oct 14, 2009
March 8th, 2020 at 3:45:37 PM permalink
Quote: OnceDear

I'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)
OnceDear
OnceDear
  • Threads: 63
  • Posts: 7471
Joined: Jun 1, 2014
March 8th, 2020 at 4:49:04 PM permalink
Quote: Wizard

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 = (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
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.
ssho88
ssho88
  • Threads: 53
  • Posts: 657
Joined: Oct 16, 2011
March 8th, 2020 at 6:30:56 PM permalink
P(z=even) = 0.4646 ?
Wizard
Administrator
Wizard
  • Threads: 1491
  • Posts: 26435
Joined: Oct 14, 2009
March 8th, 2020 at 6:37:36 PM permalink
Quote: OnceDear

I 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)
Wizard
Administrator
Wizard
  • Threads: 1491
  • Posts: 26435
Joined: Oct 14, 2009
March 8th, 2020 at 6:38:42 PM permalink
Quote: ssho88

P(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)
ssho88
ssho88
  • Threads: 53
  • Posts: 657
Joined: Oct 16, 2011
March 8th, 2020 at 9:02:53 PM permalink
Can't show the step by step solution, just pure simulation results. LOL
CrystalMath
CrystalMath
  • Threads: 8
  • Posts: 1909
Joined: May 10, 2011
March 8th, 2020 at 9:04:08 PM permalink

Fun problem.
I get (5-pi)/4

Not trying to win a beer, just having fun.
I heart Crystal Math.
Wizard
Administrator
Wizard
  • Threads: 1491
  • Posts: 26435
Joined: Oct 14, 2009
March 8th, 2020 at 9:24:56 PM permalink
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.

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Wizard
Administrator
Wizard
  • Threads: 1491
  • Posts: 26435
Joined: Oct 14, 2009
March 9th, 2020 at 7:26:20 AM permalink
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)
ChesterDog
ChesterDog
  • Threads: 8
  • Posts: 1480
Joined: Jul 26, 2010
March 9th, 2020 at 11:42:20 AM permalink
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.
gordonm888
Administrator
gordonm888
  • Threads: 60
  • Posts: 5005
Joined: Feb 18, 2015
March 9th, 2020 at 2:04:21 PM permalink
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.
Wizard
Administrator
Wizard
  • Threads: 1491
  • Posts: 26435
Joined: Oct 14, 2009
March 9th, 2020 at 5:40:06 PM permalink
Quote: gordonm888

I 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)
Klopp
Klopp
  • Threads: 7
  • Posts: 16
Joined: Mar 10, 2020
March 11th, 2020 at 12:28:24 AM permalink
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
  • Jump to: