Where have I gone wrong in this four door Monty Hall Problem?Monty Hall Three-Door PuzzleMonty hall problem extended.Monty Hall Problem IntuitionMonty Hall Problem - Toss a coin to decide if I switch the door.Four-door Monty Hall problem, but two doors are opened, one by one.Monty Hall Problem with twistMonty Hall problem with 7 doorsMonty Hall problem with biased door selection probabilityMonty Hall problem generalized to $n$ doorsHow does the Monty Hall Problem work?
Does an unused member variable take up memory?
Virginia employer terminated employee and wants signing bonus returned
What do *foreign films* mean for an American?
Can't make sense of a paragraph from Lovecraft
What is better: yes / no radio, or simple checkbox?
Why do we say ‘pairwise disjoint’, rather than ‘disjoint’?
Why is a very small peak with larger m/z not considered to be the molecular ion?
Power Strip for Europe
Making a kiddush for a girl that has hard time finding shidduch
When Schnorr signatures are part of Bitcoin will it be possible validate each block with only one signature validation?
What is Tony Stark injecting into himself in Iron Man 3?
Why does Solve lock up when trying to solve the quadratic equation with large integers?
Specifying a starting column with colortbl package and xcolor
Dynamic Linkage of LocatorPane and InputField
Why do phishing e-mails use faked e-mail addresses instead of the real one?
Proving a statement about real numbers
After `ssh` without `-X` to a machine, is it possible to change `$DISPLAY` to make it work like `ssh -X`?
What is the population of Romulus in the TNG era?
How exactly does an Ethernet collision happen in the cable, since nodes use different circuits for Tx and Rx?
How do we create new idioms and use them in a novel?
Was it really inappropriate to write a pull request for the company I interviewed with?
What would be the most expensive material to an intergalactic society?
Why couldn't the separatists legally leave the Republic?
I reported the illegal activity of my boss to his boss. My boss found out. Now I am being punished. What should I do?
Where have I gone wrong in this four door Monty Hall Problem?
Monty Hall Three-Door PuzzleMonty hall problem extended.Monty Hall Problem IntuitionMonty Hall Problem - Toss a coin to decide if I switch the door.Four-door Monty Hall problem, but two doors are opened, one by one.Monty Hall Problem with twistMonty Hall problem with 7 doorsMonty Hall problem with biased door selection probabilityMonty Hall problem generalized to $n$ doorsHow does the Monty Hall Problem work?
$begingroup$
Consider the classic Monty Hall Problem, but with four doors, labelled A, B, C, and D.
I want to calculate the probability of winning with the strategy "choose, switch, stick". I'm struggling to get from maths I think is right to the agreed upon answer, so was hoping someone could help.
I start off by picking door A. Door B is opened, so my probability distribution is as follows:
A(1/4), B(0), C(3/8), D(3/8)
This makes sense as I learnt nothing about door A, and door C and D share the remaining probability.
I then switch arbitrarily to C (C and D are indifferent). At this point, Monty will either open door A or door D.
If Monty opens door A, I propose the following distribution:
A(0), B(0), C(1/2), D(1/2)
On the other hand, if Monty opens door D:
A(4/7), B(0), C(3/7), D(0)
To calculate the probability of my strategy succeeding, I then need to combine the probability of winning depending on which door was opened.
From my understanding, this should be 3/8: http://mathworld.wolfram.com/MontyHallProblem.html. However, I haven't reached this answer regardless of how I try to combine the two scenarios.
probability monty-hall
$endgroup$
add a comment |
$begingroup$
Consider the classic Monty Hall Problem, but with four doors, labelled A, B, C, and D.
I want to calculate the probability of winning with the strategy "choose, switch, stick". I'm struggling to get from maths I think is right to the agreed upon answer, so was hoping someone could help.
I start off by picking door A. Door B is opened, so my probability distribution is as follows:
A(1/4), B(0), C(3/8), D(3/8)
This makes sense as I learnt nothing about door A, and door C and D share the remaining probability.
I then switch arbitrarily to C (C and D are indifferent). At this point, Monty will either open door A or door D.
If Monty opens door A, I propose the following distribution:
A(0), B(0), C(1/2), D(1/2)
On the other hand, if Monty opens door D:
A(4/7), B(0), C(3/7), D(0)
To calculate the probability of my strategy succeeding, I then need to combine the probability of winning depending on which door was opened.
From my understanding, this should be 3/8: http://mathworld.wolfram.com/MontyHallProblem.html. However, I haven't reached this answer regardless of how I try to combine the two scenarios.
probability monty-hall
$endgroup$
$begingroup$
Is there any rational to when or if Monty will show you your original door? that aspect if very different from the original monty hall problem. And what does Monty do if you choose not to switch?
$endgroup$
– fleablood
Mar 7 at 0:22
add a comment |
$begingroup$
Consider the classic Monty Hall Problem, but with four doors, labelled A, B, C, and D.
I want to calculate the probability of winning with the strategy "choose, switch, stick". I'm struggling to get from maths I think is right to the agreed upon answer, so was hoping someone could help.
I start off by picking door A. Door B is opened, so my probability distribution is as follows:
A(1/4), B(0), C(3/8), D(3/8)
This makes sense as I learnt nothing about door A, and door C and D share the remaining probability.
I then switch arbitrarily to C (C and D are indifferent). At this point, Monty will either open door A or door D.
If Monty opens door A, I propose the following distribution:
A(0), B(0), C(1/2), D(1/2)
On the other hand, if Monty opens door D:
A(4/7), B(0), C(3/7), D(0)
To calculate the probability of my strategy succeeding, I then need to combine the probability of winning depending on which door was opened.
From my understanding, this should be 3/8: http://mathworld.wolfram.com/MontyHallProblem.html. However, I haven't reached this answer regardless of how I try to combine the two scenarios.
probability monty-hall
$endgroup$
Consider the classic Monty Hall Problem, but with four doors, labelled A, B, C, and D.
I want to calculate the probability of winning with the strategy "choose, switch, stick". I'm struggling to get from maths I think is right to the agreed upon answer, so was hoping someone could help.
I start off by picking door A. Door B is opened, so my probability distribution is as follows:
A(1/4), B(0), C(3/8), D(3/8)
This makes sense as I learnt nothing about door A, and door C and D share the remaining probability.
I then switch arbitrarily to C (C and D are indifferent). At this point, Monty will either open door A or door D.
If Monty opens door A, I propose the following distribution:
A(0), B(0), C(1/2), D(1/2)
On the other hand, if Monty opens door D:
A(4/7), B(0), C(3/7), D(0)
To calculate the probability of my strategy succeeding, I then need to combine the probability of winning depending on which door was opened.
From my understanding, this should be 3/8: http://mathworld.wolfram.com/MontyHallProblem.html. However, I haven't reached this answer regardless of how I try to combine the two scenarios.
probability monty-hall
probability monty-hall
asked Mar 6 at 16:42
Oliver DunkOliver Dunk
213
213
$begingroup$
Is there any rational to when or if Monty will show you your original door? that aspect if very different from the original monty hall problem. And what does Monty do if you choose not to switch?
$endgroup$
– fleablood
Mar 7 at 0:22
add a comment |
$begingroup$
Is there any rational to when or if Monty will show you your original door? that aspect if very different from the original monty hall problem. And what does Monty do if you choose not to switch?
$endgroup$
– fleablood
Mar 7 at 0:22
$begingroup$
Is there any rational to when or if Monty will show you your original door? that aspect if very different from the original monty hall problem. And what does Monty do if you choose not to switch?
$endgroup$
– fleablood
Mar 7 at 0:22
$begingroup$
Is there any rational to when or if Monty will show you your original door? that aspect if very different from the original monty hall problem. And what does Monty do if you choose not to switch?
$endgroup$
– fleablood
Mar 7 at 0:22
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
When Monty opens the second door after you've switched to C, the conditional probability that the prize is behind door C (viz. $ frac38 $) does not change, for the same reason that it doesn't change for the door you've picked in the original Monty Hall problem. Thus, the probability that the prize is behind the remaining unopened door is $ frac58 $, regardless of whether that door is door A or door D.
Addendum: In light of Misha Lavrov's answer I should point out that in my above comment I have made the standard assumption that when Monty chooses a door to open he always picks one equiprobably from among the unselected that conceal booby prizes, and independently of any other occurrence
$endgroup$
add a comment |
$begingroup$
We need to assume something about Monty's strategy for opening doors. Let's assume that he opens a randomly chosen door of all doors that you haven't picked without the prize behind it, as is traditional.
Case 1: Monty opens B, then D.
Suppose the prize is behind door A. The probability that Monty opens door B when you pick door A, then opens door D when you switch to door C, is $frac13 cdot 1 = frac13$.
Suppose the prize is behind door C. The probability that Monty opens door B when you pick door A, then opens door D when you switch to door C, is $frac12 cdot frac12 = frac14$.
Initially, doors A and C were equally likely, but Monty's actions are $frac43$ times likelier if the prize is behind door A. So the probability is $frac47$ for door A and $frac37$ for door C (the odds are $frac43 : 1$).
Case 2: Monty opens B, then A.
Suppose the prize is behind door C. The probability that Monty opens door B when you pick door A, then opens door A when you switch to door C, is $frac12 cdot frac12 = frac14$ (as before).
Suppose the prize is behind door D. The probability that Monty opens door B when you pick door A, then opens door A when you switch to door C, is $frac12 cdot 1 = frac12$.
Initially, doors C and D were equally likely, but Monty's actions are $2$ times likelier if the prize is behind door D. So the probability is $frac13$ for door C and $frac23$ for door D (the odds are $1 : 2$).
Overall probability
But the above calculations are irrelevant for the problem you want to solve.
To figure out how good your chances are if you plan to pick a door, switch after the first opened door, and then stick with that choice no matter what, you don't need to do casework based on the second opened door, since it won't affect your decision. You've computed probabilities of $frac14, frac38, frac38$ for doors A, C, and D, so the final answer if you switch to door C is $frac38$.
$endgroup$
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',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
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
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3137793%2fwhere-have-i-gone-wrong-in-this-four-door-monty-hall-problem%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
When Monty opens the second door after you've switched to C, the conditional probability that the prize is behind door C (viz. $ frac38 $) does not change, for the same reason that it doesn't change for the door you've picked in the original Monty Hall problem. Thus, the probability that the prize is behind the remaining unopened door is $ frac58 $, regardless of whether that door is door A or door D.
Addendum: In light of Misha Lavrov's answer I should point out that in my above comment I have made the standard assumption that when Monty chooses a door to open he always picks one equiprobably from among the unselected that conceal booby prizes, and independently of any other occurrence
$endgroup$
add a comment |
$begingroup$
When Monty opens the second door after you've switched to C, the conditional probability that the prize is behind door C (viz. $ frac38 $) does not change, for the same reason that it doesn't change for the door you've picked in the original Monty Hall problem. Thus, the probability that the prize is behind the remaining unopened door is $ frac58 $, regardless of whether that door is door A or door D.
Addendum: In light of Misha Lavrov's answer I should point out that in my above comment I have made the standard assumption that when Monty chooses a door to open he always picks one equiprobably from among the unselected that conceal booby prizes, and independently of any other occurrence
$endgroup$
add a comment |
$begingroup$
When Monty opens the second door after you've switched to C, the conditional probability that the prize is behind door C (viz. $ frac38 $) does not change, for the same reason that it doesn't change for the door you've picked in the original Monty Hall problem. Thus, the probability that the prize is behind the remaining unopened door is $ frac58 $, regardless of whether that door is door A or door D.
Addendum: In light of Misha Lavrov's answer I should point out that in my above comment I have made the standard assumption that when Monty chooses a door to open he always picks one equiprobably from among the unselected that conceal booby prizes, and independently of any other occurrence
$endgroup$
When Monty opens the second door after you've switched to C, the conditional probability that the prize is behind door C (viz. $ frac38 $) does not change, for the same reason that it doesn't change for the door you've picked in the original Monty Hall problem. Thus, the probability that the prize is behind the remaining unopened door is $ frac58 $, regardless of whether that door is door A or door D.
Addendum: In light of Misha Lavrov's answer I should point out that in my above comment I have made the standard assumption that when Monty chooses a door to open he always picks one equiprobably from among the unselected that conceal booby prizes, and independently of any other occurrence
edited yesterday
answered Mar 7 at 0:16
lonza leggieralonza leggiera
1,03228
1,03228
add a comment |
add a comment |
$begingroup$
We need to assume something about Monty's strategy for opening doors. Let's assume that he opens a randomly chosen door of all doors that you haven't picked without the prize behind it, as is traditional.
Case 1: Monty opens B, then D.
Suppose the prize is behind door A. The probability that Monty opens door B when you pick door A, then opens door D when you switch to door C, is $frac13 cdot 1 = frac13$.
Suppose the prize is behind door C. The probability that Monty opens door B when you pick door A, then opens door D when you switch to door C, is $frac12 cdot frac12 = frac14$.
Initially, doors A and C were equally likely, but Monty's actions are $frac43$ times likelier if the prize is behind door A. So the probability is $frac47$ for door A and $frac37$ for door C (the odds are $frac43 : 1$).
Case 2: Monty opens B, then A.
Suppose the prize is behind door C. The probability that Monty opens door B when you pick door A, then opens door A when you switch to door C, is $frac12 cdot frac12 = frac14$ (as before).
Suppose the prize is behind door D. The probability that Monty opens door B when you pick door A, then opens door A when you switch to door C, is $frac12 cdot 1 = frac12$.
Initially, doors C and D were equally likely, but Monty's actions are $2$ times likelier if the prize is behind door D. So the probability is $frac13$ for door C and $frac23$ for door D (the odds are $1 : 2$).
Overall probability
But the above calculations are irrelevant for the problem you want to solve.
To figure out how good your chances are if you plan to pick a door, switch after the first opened door, and then stick with that choice no matter what, you don't need to do casework based on the second opened door, since it won't affect your decision. You've computed probabilities of $frac14, frac38, frac38$ for doors A, C, and D, so the final answer if you switch to door C is $frac38$.
$endgroup$
add a comment |
$begingroup$
We need to assume something about Monty's strategy for opening doors. Let's assume that he opens a randomly chosen door of all doors that you haven't picked without the prize behind it, as is traditional.
Case 1: Monty opens B, then D.
Suppose the prize is behind door A. The probability that Monty opens door B when you pick door A, then opens door D when you switch to door C, is $frac13 cdot 1 = frac13$.
Suppose the prize is behind door C. The probability that Monty opens door B when you pick door A, then opens door D when you switch to door C, is $frac12 cdot frac12 = frac14$.
Initially, doors A and C were equally likely, but Monty's actions are $frac43$ times likelier if the prize is behind door A. So the probability is $frac47$ for door A and $frac37$ for door C (the odds are $frac43 : 1$).
Case 2: Monty opens B, then A.
Suppose the prize is behind door C. The probability that Monty opens door B when you pick door A, then opens door A when you switch to door C, is $frac12 cdot frac12 = frac14$ (as before).
Suppose the prize is behind door D. The probability that Monty opens door B when you pick door A, then opens door A when you switch to door C, is $frac12 cdot 1 = frac12$.
Initially, doors C and D were equally likely, but Monty's actions are $2$ times likelier if the prize is behind door D. So the probability is $frac13$ for door C and $frac23$ for door D (the odds are $1 : 2$).
Overall probability
But the above calculations are irrelevant for the problem you want to solve.
To figure out how good your chances are if you plan to pick a door, switch after the first opened door, and then stick with that choice no matter what, you don't need to do casework based on the second opened door, since it won't affect your decision. You've computed probabilities of $frac14, frac38, frac38$ for doors A, C, and D, so the final answer if you switch to door C is $frac38$.
$endgroup$
add a comment |
$begingroup$
We need to assume something about Monty's strategy for opening doors. Let's assume that he opens a randomly chosen door of all doors that you haven't picked without the prize behind it, as is traditional.
Case 1: Monty opens B, then D.
Suppose the prize is behind door A. The probability that Monty opens door B when you pick door A, then opens door D when you switch to door C, is $frac13 cdot 1 = frac13$.
Suppose the prize is behind door C. The probability that Monty opens door B when you pick door A, then opens door D when you switch to door C, is $frac12 cdot frac12 = frac14$.
Initially, doors A and C were equally likely, but Monty's actions are $frac43$ times likelier if the prize is behind door A. So the probability is $frac47$ for door A and $frac37$ for door C (the odds are $frac43 : 1$).
Case 2: Monty opens B, then A.
Suppose the prize is behind door C. The probability that Monty opens door B when you pick door A, then opens door A when you switch to door C, is $frac12 cdot frac12 = frac14$ (as before).
Suppose the prize is behind door D. The probability that Monty opens door B when you pick door A, then opens door A when you switch to door C, is $frac12 cdot 1 = frac12$.
Initially, doors C and D were equally likely, but Monty's actions are $2$ times likelier if the prize is behind door D. So the probability is $frac13$ for door C and $frac23$ for door D (the odds are $1 : 2$).
Overall probability
But the above calculations are irrelevant for the problem you want to solve.
To figure out how good your chances are if you plan to pick a door, switch after the first opened door, and then stick with that choice no matter what, you don't need to do casework based on the second opened door, since it won't affect your decision. You've computed probabilities of $frac14, frac38, frac38$ for doors A, C, and D, so the final answer if you switch to door C is $frac38$.
$endgroup$
We need to assume something about Monty's strategy for opening doors. Let's assume that he opens a randomly chosen door of all doors that you haven't picked without the prize behind it, as is traditional.
Case 1: Monty opens B, then D.
Suppose the prize is behind door A. The probability that Monty opens door B when you pick door A, then opens door D when you switch to door C, is $frac13 cdot 1 = frac13$.
Suppose the prize is behind door C. The probability that Monty opens door B when you pick door A, then opens door D when you switch to door C, is $frac12 cdot frac12 = frac14$.
Initially, doors A and C were equally likely, but Monty's actions are $frac43$ times likelier if the prize is behind door A. So the probability is $frac47$ for door A and $frac37$ for door C (the odds are $frac43 : 1$).
Case 2: Monty opens B, then A.
Suppose the prize is behind door C. The probability that Monty opens door B when you pick door A, then opens door A when you switch to door C, is $frac12 cdot frac12 = frac14$ (as before).
Suppose the prize is behind door D. The probability that Monty opens door B when you pick door A, then opens door A when you switch to door C, is $frac12 cdot 1 = frac12$.
Initially, doors C and D were equally likely, but Monty's actions are $2$ times likelier if the prize is behind door D. So the probability is $frac13$ for door C and $frac23$ for door D (the odds are $1 : 2$).
Overall probability
But the above calculations are irrelevant for the problem you want to solve.
To figure out how good your chances are if you plan to pick a door, switch after the first opened door, and then stick with that choice no matter what, you don't need to do casework based on the second opened door, since it won't affect your decision. You've computed probabilities of $frac14, frac38, frac38$ for doors A, C, and D, so the final answer if you switch to door C is $frac38$.
edited Mar 7 at 17:47
answered Mar 7 at 0:27
Misha LavrovMisha Lavrov
47.4k657107
47.4k657107
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3137793%2fwhere-have-i-gone-wrong-in-this-four-door-monty-hall-problem%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
Is there any rational to when or if Monty will show you your original door? that aspect if very different from the original monty hall problem. And what does Monty do if you choose not to switch?
$endgroup$
– fleablood
Mar 7 at 0:22