Task on mix of calculus with probably probability

Clash Royale CLAN TAG#URR8PPP

up vote
0
down vote

favorite

I have this task to solve:

A father suggests two algorithms to divide a round pie between his sons: A) The elder son gets 2/3, and the younger son gets 1/3; B) The pie is cut along the line passing through two points chosen at random at its circumference, and the younger son gets the smaller piece. Which algorithm gives a larger mathematical expectation of the younger son's part?

EDIT: I have a brutal error below. My only question is:

What is the name of this class of problems? What topic should I read to be able to solve these?

To model algorithm B):

Let's take just half a circle and put it on a graph in such a way that the first randomly chosen point lies exactly at (0,0). Let radius $r = 1$

Circle and a line dividing it

Then let's choose another random point and define the equations for these functions:

$y_l = kx$ for the green line, $b = 0$ because $y_l(0) = 0$

$y_c = sqrt1 - (x - 1)^2, y > 0, x > 0$ for the red line-circumference,
where the first 1 is radius

The $S$ area on the graph is equal to the part which will be given to the younger brother.

Then the area of $S$ must be equal

EDIT: the error is in subtracting $int_0^2 y_l mathrmdx$
as this subtracts extra area in the top-right corner

beginalign*
S &= int_0^2 y_c mathrmdx - int_0^2 y_l mathrmdx \
& = int_0^2 sqrt1 - (x - 1)^2 mathrmdx - int_0^2 kx mathrmdx \
& textlet's substitute ,
x - 1 = sin(u) :textand: mathrmdx = cos(u)mathrmdu :
textfor 1st eq. and keep int. intervals as a and b for simplicity \
& = int_a^b sqrt1 - sin^2(u):cos(u)mathrmdu - (frackx^22)|_0^2 \
& = int_a^b cos^2(u)mathrmdu - (frackx^22)|_0^2 \
& = int_b^b (frac12 + fraccos(2u)2) mathrmdu -
(frackx^22)|_0^2 \
& = (frac12u - fracsin(2u)4)|_a^b - (frackx^22)|_0^2 \
& mid:sin(2u) = 2sin(u)cos(u) = 2sin(u)sqrt1 - sin^2(u):mid \
& = (fracarcsin(x-1)2 - frac(x - 1)sqrt1-(x-1)^22)|_0^2
- (frackx^22)|_0^2 \
& = [(fracpi4 - 0) - (frac-pi4 - 0)] - (2k -0) \
& = fracpi2 - 2k
endalign*

So I have that in the algorithm A) the younger son gets 1/3 of a circle,
$S_c = pi times 1^2 = pi$, so it's $fracpi3$.

In the algorithm B) the younger son gets $S(k) = fracpi2 - 2k$.

And the question is: Which algorithm gives a larger mathematical expectation of the younger son's part?

I have 2 questions:

1. Where do I go from there?




2. What is the name of this class of problems? What topic should I read to be
   able to solve these?

calculus probability reference-request terminology book-recommendation

share|cite|improve this question

edited Jul 17 at 5:08

joriki

164k10180328

asked Jul 16 at 19:08

Dmitry

82

• 
  
  
  

  
  

  
  
  
  
  
  

  
  You wrote: "Which algorithm gives a larger mathematical expectation of the younger son's part?" Are you sure this is the question? Or is the question possibly, "Is the second algorithm more likely to give the younger son more than $1/3$ of the pie, or less?" ETA: It could be either. I just want to be sure that we're answering the right question.
  – Brian Tung
  Jul 16 at 20:36
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @BrianTung yes, I have just checked it. Though your proposed question seems as a better restatement of the original question, I also had this thought in mind.
  – Dmitry
  Jul 16 at 20:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  might be helpful everipedia.org/wiki/Bertrand_paradox_(probability)
  – Jonas Kgomo
  Jul 17 at 3:52
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Exact same problem ,different wording math.stackexchange.com/questions/1011234/…
  – Jonas Kgomo
  Jul 17 at 4:10
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @Jonas12 thank you for taking your time and finding similar problem! Bertrand paradox is interesting as well!
  – Dmitry
  Jul 17 at 22:16
  

  
  
  
  

 | 
show 3 more comments

up vote
0
down vote

favorite

I have this task to solve:

A father suggests two algorithms to divide a round pie between his sons: A) The elder son gets 2/3, and the younger son gets 1/3; B) The pie is cut along the line passing through two points chosen at random at its circumference, and the younger son gets the smaller piece. Which algorithm gives a larger mathematical expectation of the younger son's part?

EDIT: I have a brutal error below. My only question is:

What is the name of this class of problems? What topic should I read to be able to solve these?

To model algorithm B):

Let's take just half a circle and put it on a graph in such a way that the first randomly chosen point lies exactly at (0,0). Let radius $r = 1$

Circle and a line dividing it

Then let's choose another random point and define the equations for these functions:

$y_l = kx$ for the green line, $b = 0$ because $y_l(0) = 0$

$y_c = sqrt1 - (x - 1)^2, y > 0, x > 0$ for the red line-circumference,
where the first 1 is radius

The $S$ area on the graph is equal to the part which will be given to the younger brother.

Then the area of $S$ must be equal

EDIT: the error is in subtracting $int_0^2 y_l mathrmdx$
as this subtracts extra area in the top-right corner

beginalign*
S &= int_0^2 y_c mathrmdx - int_0^2 y_l mathrmdx \
& = int_0^2 sqrt1 - (x - 1)^2 mathrmdx - int_0^2 kx mathrmdx \
& textlet's substitute ,
x - 1 = sin(u) :textand: mathrmdx = cos(u)mathrmdu :
textfor 1st eq. and keep int. intervals as a and b for simplicity \
& = int_a^b sqrt1 - sin^2(u):cos(u)mathrmdu - (frackx^22)|_0^2 \
& = int_a^b cos^2(u)mathrmdu - (frackx^22)|_0^2 \
& = int_b^b (frac12 + fraccos(2u)2) mathrmdu -
(frackx^22)|_0^2 \
& = (frac12u - fracsin(2u)4)|_a^b - (frackx^22)|_0^2 \
& mid:sin(2u) = 2sin(u)cos(u) = 2sin(u)sqrt1 - sin^2(u):mid \
& = (fracarcsin(x-1)2 - frac(x - 1)sqrt1-(x-1)^22)|_0^2
- (frackx^22)|_0^2 \
& = [(fracpi4 - 0) - (frac-pi4 - 0)] - (2k -0) \
& = fracpi2 - 2k
endalign*

So I have that in the algorithm A) the younger son gets 1/3 of a circle,
$S_c = pi times 1^2 = pi$, so it's $fracpi3$.

In the algorithm B) the younger son gets $S(k) = fracpi2 - 2k$.

And the question is: Which algorithm gives a larger mathematical expectation of the younger son's part?

I have 2 questions:

1. Where do I go from there?




2. What is the name of this class of problems? What topic should I read to be
   able to solve these?

calculus probability reference-request terminology book-recommendation

share|cite|improve this question

edited Jul 17 at 5:08

joriki

164k10180328

asked Jul 16 at 19:08

Dmitry

82

• 
  
  
  

  
  

  
  
  
  
  
  

  
  You wrote: "Which algorithm gives a larger mathematical expectation of the younger son's part?" Are you sure this is the question? Or is the question possibly, "Is the second algorithm more likely to give the younger son more than $1/3$ of the pie, or less?" ETA: It could be either. I just want to be sure that we're answering the right question.
  – Brian Tung
  Jul 16 at 20:36
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @BrianTung yes, I have just checked it. Though your proposed question seems as a better restatement of the original question, I also had this thought in mind.
  – Dmitry
  Jul 16 at 20:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  might be helpful everipedia.org/wiki/Bertrand_paradox_(probability)
  – Jonas Kgomo
  Jul 17 at 3:52
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Exact same problem ,different wording math.stackexchange.com/questions/1011234/…
  – Jonas Kgomo
  Jul 17 at 4:10
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @Jonas12 thank you for taking your time and finding similar problem! Bertrand paradox is interesting as well!
  – Dmitry
  Jul 17 at 22:16
  

  
  
  
  

 | 
show 3 more comments

up vote
0
down vote

favorite

up vote
0
down vote

favorite

I have this task to solve:

A father suggests two algorithms to divide a round pie between his sons: A) The elder son gets 2/3, and the younger son gets 1/3; B) The pie is cut along the line passing through two points chosen at random at its circumference, and the younger son gets the smaller piece. Which algorithm gives a larger mathematical expectation of the younger son's part?

EDIT: I have a brutal error below. My only question is:

What is the name of this class of problems? What topic should I read to be able to solve these?

To model algorithm B):

Let's take just half a circle and put it on a graph in such a way that the first randomly chosen point lies exactly at (0,0). Let radius $r = 1$

Circle and a line dividing it

Then let's choose another random point and define the equations for these functions:

$y_l = kx$ for the green line, $b = 0$ because $y_l(0) = 0$

$y_c = sqrt1 - (x - 1)^2, y > 0, x > 0$ for the red line-circumference,
where the first 1 is radius

The $S$ area on the graph is equal to the part which will be given to the younger brother.

Then the area of $S$ must be equal

EDIT: the error is in subtracting $int_0^2 y_l mathrmdx$
as this subtracts extra area in the top-right corner

beginalign*
S &= int_0^2 y_c mathrmdx - int_0^2 y_l mathrmdx \
& = int_0^2 sqrt1 - (x - 1)^2 mathrmdx - int_0^2 kx mathrmdx \
& textlet's substitute ,
x - 1 = sin(u) :textand: mathrmdx = cos(u)mathrmdu :
textfor 1st eq. and keep int. intervals as a and b for simplicity \
& = int_a^b sqrt1 - sin^2(u):cos(u)mathrmdu - (frackx^22)|_0^2 \
& = int_a^b cos^2(u)mathrmdu - (frackx^22)|_0^2 \
& = int_b^b (frac12 + fraccos(2u)2) mathrmdu -
(frackx^22)|_0^2 \
& = (frac12u - fracsin(2u)4)|_a^b - (frackx^22)|_0^2 \
& mid:sin(2u) = 2sin(u)cos(u) = 2sin(u)sqrt1 - sin^2(u):mid \
& = (fracarcsin(x-1)2 - frac(x - 1)sqrt1-(x-1)^22)|_0^2
- (frackx^22)|_0^2 \
& = [(fracpi4 - 0) - (frac-pi4 - 0)] - (2k -0) \
& = fracpi2 - 2k
endalign*

So I have that in the algorithm A) the younger son gets 1/3 of a circle,
$S_c = pi times 1^2 = pi$, so it's $fracpi3$.

In the algorithm B) the younger son gets $S(k) = fracpi2 - 2k$.

And the question is: Which algorithm gives a larger mathematical expectation of the younger son's part?

I have 2 questions:

1. Where do I go from there?




2. What is the name of this class of problems? What topic should I read to be
   able to solve these?

calculus probability reference-request terminology book-recommendation

share|cite|improve this question

edited Jul 17 at 5:08

joriki

164k10180328

asked Jul 16 at 19:08

Dmitry

82

I have this task to solve:

A father suggests two algorithms to divide a round pie between his sons: A) The elder son gets 2/3, and the younger son gets 1/3; B) The pie is cut along the line passing through two points chosen at random at its circumference, and the younger son gets the smaller piece. Which algorithm gives a larger mathematical expectation of the younger son's part?

EDIT: I have a brutal error below. My only question is:

What is the name of this class of problems? What topic should I read to be able to solve these?

To model algorithm B):

Let's take just half a circle and put it on a graph in such a way that the first randomly chosen point lies exactly at (0,0). Let radius $r = 1$

Circle and a line dividing it

Then let's choose another random point and define the equations for these functions:

$y_l = kx$ for the green line, $b = 0$ because $y_l(0) = 0$

$y_c = sqrt1 - (x - 1)^2, y > 0, x > 0$ for the red line-circumference,
where the first 1 is radius

The $S$ area on the graph is equal to the part which will be given to the younger brother.

Then the area of $S$ must be equal

EDIT: the error is in subtracting $int_0^2 y_l mathrmdx$
as this subtracts extra area in the top-right corner

beginalign*
S &= int_0^2 y_c mathrmdx - int_0^2 y_l mathrmdx \
& = int_0^2 sqrt1 - (x - 1)^2 mathrmdx - int_0^2 kx mathrmdx \
& textlet's substitute ,
x - 1 = sin(u) :textand: mathrmdx = cos(u)mathrmdu :
textfor 1st eq. and keep int. intervals as a and b for simplicity \
& = int_a^b sqrt1 - sin^2(u):cos(u)mathrmdu - (frackx^22)|_0^2 \
& = int_a^b cos^2(u)mathrmdu - (frackx^22)|_0^2 \
& = int_b^b (frac12 + fraccos(2u)2) mathrmdu -
(frackx^22)|_0^2 \
& = (frac12u - fracsin(2u)4)|_a^b - (frackx^22)|_0^2 \
& mid:sin(2u) = 2sin(u)cos(u) = 2sin(u)sqrt1 - sin^2(u):mid \
& = (fracarcsin(x-1)2 - frac(x - 1)sqrt1-(x-1)^22)|_0^2
- (frackx^22)|_0^2 \
& = [(fracpi4 - 0) - (frac-pi4 - 0)] - (2k -0) \
& = fracpi2 - 2k
endalign*

So I have that in the algorithm A) the younger son gets 1/3 of a circle,
$S_c = pi times 1^2 = pi$, so it's $fracpi3$.

In the algorithm B) the younger son gets $S(k) = fracpi2 - 2k$.

And the question is: Which algorithm gives a larger mathematical expectation of the younger son's part?

I have 2 questions:

1. Where do I go from there?




2. What is the name of this class of problems? What topic should I read to be
   able to solve these?

calculus probability reference-request terminology book-recommendation

share|cite|improve this question

edited Jul 17 at 5:08

joriki

164k10180328

asked Jul 16 at 19:08

Dmitry

82

share|cite|improve this question

share|cite|improve this question

edited Jul 17 at 5:08

joriki

164k10180328

edited Jul 17 at 5:08

joriki

164k10180328

edited Jul 17 at 5:08

joriki

164k10180328

164k10180328

asked Jul 16 at 19:08

Dmitry

82

asked Jul 16 at 19:08

Dmitry

82

asked Jul 16 at 19:08

Dmitry

82

82

• 
  
  
  

  
  

  
  
  
  
  
  

  
  You wrote: "Which algorithm gives a larger mathematical expectation of the younger son's part?" Are you sure this is the question? Or is the question possibly, "Is the second algorithm more likely to give the younger son more than $1/3$ of the pie, or less?" ETA: It could be either. I just want to be sure that we're answering the right question.
  – Brian Tung
  Jul 16 at 20:36
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @BrianTung yes, I have just checked it. Though your proposed question seems as a better restatement of the original question, I also had this thought in mind.
  – Dmitry
  Jul 16 at 20:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  might be helpful everipedia.org/wiki/Bertrand_paradox_(probability)
  – Jonas Kgomo
  Jul 17 at 3:52
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Exact same problem ,different wording math.stackexchange.com/questions/1011234/…
  – Jonas Kgomo
  Jul 17 at 4:10
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @Jonas12 thank you for taking your time and finding similar problem! Bertrand paradox is interesting as well!
  – Dmitry
  Jul 17 at 22:16
  

  
  
  
  

 | 
show 3 more comments

• 
  
  
  

  
  

  
  
  
  
  
  

  
  You wrote: "Which algorithm gives a larger mathematical expectation of the younger son's part?" Are you sure this is the question? Or is the question possibly, "Is the second algorithm more likely to give the younger son more than $1/3$ of the pie, or less?" ETA: It could be either. I just want to be sure that we're answering the right question.
  – Brian Tung
  Jul 16 at 20:36
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @BrianTung yes, I have just checked it. Though your proposed question seems as a better restatement of the original question, I also had this thought in mind.
  – Dmitry
  Jul 16 at 20:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  might be helpful everipedia.org/wiki/Bertrand_paradox_(probability)
  – Jonas Kgomo
  Jul 17 at 3:52
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Exact same problem ,different wording math.stackexchange.com/questions/1011234/…
  – Jonas Kgomo
  Jul 17 at 4:10
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @Jonas12 thank you for taking your time and finding similar problem! Bertrand paradox is interesting as well!
  – Dmitry
  Jul 17 at 22:16
  

  
  
  
  

You wrote: "Which algorithm gives a larger mathematical expectation of the younger son's part?" Are you sure this is the question? Or is the question possibly, "Is the second algorithm more likely to give the younger son more than $1/3$ of the pie, or less?" ETA: It could be either. I just want to be sure that we're answering the right question.
– Brian Tung
Jul 16 at 20:36

You wrote: "Which algorithm gives a larger mathematical expectation of the younger son's part?" Are you sure this is the question? Or is the question possibly, "Is the second algorithm more likely to give the younger son more than $1/3$ of the pie, or less?" ETA: It could be either. I just want to be sure that we're answering the right question.
– Brian Tung
Jul 16 at 20:36

@BrianTung yes, I have just checked it. Though your proposed question seems as a better restatement of the original question, I also had this thought in mind.
– Dmitry
Jul 16 at 20:40

@BrianTung yes, I have just checked it. Though your proposed question seems as a better restatement of the original question, I also had this thought in mind.
– Dmitry
Jul 16 at 20:40

might be helpful everipedia.org/wiki/Bertrand_paradox_(probability)
– Jonas Kgomo
Jul 17 at 3:52

might be helpful everipedia.org/wiki/Bertrand_paradox_(probability)
– Jonas Kgomo
Jul 17 at 3:52

Exact same problem ,different wording math.stackexchange.com/questions/1011234/…
– Jonas Kgomo
Jul 17 at 4:10

Exact same problem ,different wording math.stackexchange.com/questions/1011234/…
– Jonas Kgomo
Jul 17 at 4:10

1

1

@Jonas12 thank you for taking your time and finding similar problem! Bertrand paradox is interesting as well!
– Dmitry
Jul 17 at 22:16

@Jonas12 thank you for taking your time and finding similar problem! Bertrand paradox is interesting as well!
– Dmitry
Jul 17 at 22:16

 | 
show 3 more comments

1 Answer
1

active

oldest

votes

up vote
0
down vote



accepted

This is a problem of geometric probability (not to be confused with the geometric distribution). There's a geometric-probability tag here on math.SE with $348$ questions that you can look at to get ideas for such problems.

share|cite|improve this answer

answered Jul 17 at 5:11

joriki

164k10180328

add a comment | 

Your Answer

StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2853740%2ftask-on-mix-of-calculus-with-probably-probability%23new-answer', 'question_page');

);

Post as a guest

Name Email

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote



accepted

This is a problem of geometric probability (not to be confused with the geometric distribution). There's a geometric-probability tag here on math.SE with $348$ questions that you can look at to get ideas for such problems.

share|cite|improve this answer

answered Jul 17 at 5:11

joriki

164k10180328

add a comment | 

up vote
0
down vote



accepted

This is a problem of geometric probability (not to be confused with the geometric distribution). There's a geometric-probability tag here on math.SE with $348$ questions that you can look at to get ideas for such problems.

share|cite|improve this answer

answered Jul 17 at 5:11

joriki

164k10180328

add a comment | 

up vote
0
down vote



accepted

up vote
0
down vote



accepted

This is a problem of geometric probability (not to be confused with the geometric distribution). There's a geometric-probability tag here on math.SE with $348$ questions that you can look at to get ideas for such problems.

share|cite|improve this answer

answered Jul 17 at 5:11

joriki

164k10180328

This is a problem of geometric probability (not to be confused with the geometric distribution). There's a geometric-probability tag here on math.SE with $348$ questions that you can look at to get ideas for such problems.

share|cite|improve this answer

answered Jul 17 at 5:11

joriki

164k10180328

share|cite|improve this answer

share|cite|improve this answer

answered Jul 17 at 5:11

joriki

164k10180328

answered Jul 17 at 5:11

joriki

164k10180328

answered Jul 17 at 5:11

joriki

164k10180328

164k10180328

add a comment | 

add a comment | 

 

draft saved

draft discarded

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2853740%2ftask-on-mix-of-calculus-with-probably-probability%23new-answer', 'question_page');

);

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Name Email

Name

Email

Name Email

Name

Email