## Poll

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4 members have voted

Wizard
Joined: Oct 14, 2009
• Posts: 22046
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.

Notes:
* corrected. Previous formula incorrect.
Last edited by: Wizard on Mar 8, 2020
It's not whether you win or lose; it's whether or not you had a good bet.
scrooge
Joined: Nov 22, 2016
• Posts: 32
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
Joined: Jun 1, 2014
• Posts: 4463
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.
Take care out there. Spare a thought for the newly poor who were happy in their world just a few days ago, but whose whole way of life just collapsed..
Wizard
Joined: Oct 14, 2009
• Posts: 22046
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.
It's not whether you win or lose; it's whether or not you had a good bet.
OnceDear
Joined: Jun 1, 2014
• Posts: 4463
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
Take care out there. Spare a thought for the newly poor who were happy in their world just a few days ago, but whose whole way of life just collapsed..
ssho88
Joined: Oct 16, 2011
• Posts: 453
March 8th, 2020 at 6:30:56 PM permalink
P(z=even) = 0.4646 ?
Wizard
Joined: Oct 14, 2009
• Posts: 22046
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).
It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
Joined: Oct 14, 2009
• Posts: 22046
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.
It's not whether you win or lose; it's whether or not you had a good bet.
ssho88
Joined: Oct 16, 2011
• Posts: 453
March 8th, 2020 at 9:02:53 PM permalink
Can't show the step by step solution, just pure simulation results. LOL
CrystalMath
Joined: May 10, 2011