Monty hall problem probability 2/6?
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For the Monty hall problem, with 3 doors, two of which have sheep and 1 has a car. I calculated the probability of getting the car if you swap being 2/6 instead of 2/3. I have drawn this tree diagram of how I calculated it:
And from it I get that the probability of getting a car if you swap is 2/6 and if you stay it's 1/6, which is not the same as the actual answer 2/3 and 1/3. I have included the probabilities of getting sheep, as well as my overall probabilities, add up so I want to know what is wrong with my tree diagram, which resulted in this wrong answer. In my tree diagram I haven't included the host revealing a door, so could this be a factor as to why my answer is wrong?
probability monty-hall decision-trees
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add a comment |
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For the Monty hall problem, with 3 doors, two of which have sheep and 1 has a car. I calculated the probability of getting the car if you swap being 2/6 instead of 2/3. I have drawn this tree diagram of how I calculated it:
And from it I get that the probability of getting a car if you swap is 2/6 and if you stay it's 1/6, which is not the same as the actual answer 2/3 and 1/3. I have included the probabilities of getting sheep, as well as my overall probabilities, add up so I want to know what is wrong with my tree diagram, which resulted in this wrong answer. In my tree diagram I haven't included the host revealing a door, so could this be a factor as to why my answer is wrong?
probability monty-hall decision-trees
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27
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"In my tree diagram i haven't included the host revealing a door, so could this be a factor as to why my answer is wrong?" Yes.
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– John Hughes
Aug 7 '17 at 13:40
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What happens on the other 3/6?
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– Kevin
Aug 8 '17 at 13:07
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@JohnHughes Actually, the host revealing a door is taken into account with swapping-toggles-the-cariness property of the tree diagram.
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– wizzwizz4
Aug 8 '17 at 15:41
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@Kevin the other 3/6 is the probability of getting sheep
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– yt.
Aug 8 '17 at 17:35
add a comment |
$begingroup$
For the Monty hall problem, with 3 doors, two of which have sheep and 1 has a car. I calculated the probability of getting the car if you swap being 2/6 instead of 2/3. I have drawn this tree diagram of how I calculated it:
And from it I get that the probability of getting a car if you swap is 2/6 and if you stay it's 1/6, which is not the same as the actual answer 2/3 and 1/3. I have included the probabilities of getting sheep, as well as my overall probabilities, add up so I want to know what is wrong with my tree diagram, which resulted in this wrong answer. In my tree diagram I haven't included the host revealing a door, so could this be a factor as to why my answer is wrong?
probability monty-hall decision-trees
$endgroup$
For the Monty hall problem, with 3 doors, two of which have sheep and 1 has a car. I calculated the probability of getting the car if you swap being 2/6 instead of 2/3. I have drawn this tree diagram of how I calculated it:
And from it I get that the probability of getting a car if you swap is 2/6 and if you stay it's 1/6, which is not the same as the actual answer 2/3 and 1/3. I have included the probabilities of getting sheep, as well as my overall probabilities, add up so I want to know what is wrong with my tree diagram, which resulted in this wrong answer. In my tree diagram I haven't included the host revealing a door, so could this be a factor as to why my answer is wrong?
probability monty-hall decision-trees
probability monty-hall decision-trees
edited Aug 8 '17 at 17:17
Community♦
1
1
asked Aug 7 '17 at 13:30
yt.yt.
121110
121110
27
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"In my tree diagram i haven't included the host revealing a door, so could this be a factor as to why my answer is wrong?" Yes.
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– John Hughes
Aug 7 '17 at 13:40
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What happens on the other 3/6?
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– Kevin
Aug 8 '17 at 13:07
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@JohnHughes Actually, the host revealing a door is taken into account with swapping-toggles-the-cariness property of the tree diagram.
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– wizzwizz4
Aug 8 '17 at 15:41
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@Kevin the other 3/6 is the probability of getting sheep
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– yt.
Aug 8 '17 at 17:35
add a comment |
27
$begingroup$
"In my tree diagram i haven't included the host revealing a door, so could this be a factor as to why my answer is wrong?" Yes.
$endgroup$
– John Hughes
Aug 7 '17 at 13:40
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What happens on the other 3/6?
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– Kevin
Aug 8 '17 at 13:07
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@JohnHughes Actually, the host revealing a door is taken into account with swapping-toggles-the-cariness property of the tree diagram.
$endgroup$
– wizzwizz4
Aug 8 '17 at 15:41
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@Kevin the other 3/6 is the probability of getting sheep
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– yt.
Aug 8 '17 at 17:35
27
27
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"In my tree diagram i haven't included the host revealing a door, so could this be a factor as to why my answer is wrong?" Yes.
$endgroup$
– John Hughes
Aug 7 '17 at 13:40
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"In my tree diagram i haven't included the host revealing a door, so could this be a factor as to why my answer is wrong?" Yes.
$endgroup$
– John Hughes
Aug 7 '17 at 13:40
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What happens on the other 3/6?
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– Kevin
Aug 8 '17 at 13:07
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What happens on the other 3/6?
$endgroup$
– Kevin
Aug 8 '17 at 13:07
$begingroup$
@JohnHughes Actually, the host revealing a door is taken into account with swapping-toggles-the-cariness property of the tree diagram.
$endgroup$
– wizzwizz4
Aug 8 '17 at 15:41
$begingroup$
@JohnHughes Actually, the host revealing a door is taken into account with swapping-toggles-the-cariness property of the tree diagram.
$endgroup$
– wizzwizz4
Aug 8 '17 at 15:41
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@Kevin the other 3/6 is the probability of getting sheep
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– yt.
Aug 8 '17 at 17:35
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@Kevin the other 3/6 is the probability of getting sheep
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– yt.
Aug 8 '17 at 17:35
add a comment |
4 Answers
4
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oldest
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It's simply because your diagram assumes you stay with probability $1/2$. In that case the probability of switching and getting the car is indeed $2/6$, and the probability of switching and not getting the car is $1/6$.
But if you always switch, you're twice as likely to get each of these outcomes - instead of a $1/2$ probability of switching, you have probability $1$ of switching - so they become $2/3$ and $1/3$ respectively.
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add a comment |
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That indicates that instead of using a strategy, you are randomly choosing whether to swap or stay. The probability of winning a car by doing that is and unsurprising $1/2$.
$$tfrac 13tfrac 12+tfrac 23tfrac 12=tfrac 12$$
The point of the scenario is that if you use strategy of always swapping, the probability of winning a car becomes: $2/3$.
$$tfrac 13tfrac 01+tfrac 23tfrac 11=tfrac 23$$
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2
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The point is that if you switch with probability $1/2$ you become equally likely to pick either of the doors the host hasn't opened - and exactly one of them has a car behind it.
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– Especially Lime
Aug 7 '17 at 13:45
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Indeed...I wrote too soon. :(
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– John Hughes
Aug 7 '17 at 13:45
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Thank you, i get the "always swapping strategy" now but it don't quite understand the sums with two fractions next to each other and how in the top one the probability of getting a car is 1/2, could you explain this please? :)
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– yt.
Aug 7 '17 at 14:01
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It comes dirrect from your diagram, @Ruby. There are two paths on it that lead to winning a car: pick a car ($tfrac 13$) and stay ($tfrac 12$) or pick a sheep $(tfrac 23)$ and swap ($tfrac 12$).
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– Graham Kemp
Aug 7 '17 at 23:30
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@GrahamKemp Okay I see now, and then do you times the two fractions that are next to each other together, and the sum of them is 1/2? If so is it usually written like that or is there a multiplication symbol between the two fractions? Thanks
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– yt.
Aug 8 '17 at 17:56
add a comment |
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You're calculating the probability wrong. If you decide at random with equal probability to swap or stay, in $tfrac{2}{6}$ of all cases you will have swapped and gotten the car.
But that is not the probability of if you decide to swap, would you get the car?
Note that in exactly $tfrac{3}{6}=tfrac{1}{2}$ of all cases, you will have swapped, and in $tfrac{1}{2}$ of all cases you will have stayed. But look at those cases.
Of the $tfrac{3}{6}$ cases where you will have swapped, there is a $tfrac{2}{6}$ chance of having swapped and gotten a car while there is a $tfrac{1}{6}$ of having swapped and gotten a goat sheep.
On the other hand there is a $tfrac{1}{6}$ chance of having stayed and gotten a car while there is a $tfrac{2}{6}$ of having swapped and gotten a sheep.
So if you swap, the probability of getting a car is twice that of getting a sheep, while if you stay, the chance of getting a sheep is twice that of getting a car.
Which is the exact outcome of the canonical Monty Hall problem.
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The key to the solution of the Monty Hall problem is that Monty Hall reveals one of the incorrect solutions after stage one. The problem ends up being exclusively about the door that you choose versus the one that is left.
This means that the probabilities are as follows:
We will assume Door 1 is the car and Door 2 and 3 are sheep.
Choose Door 1:
Monty can reveal door 2 or 3, so you have a choice between 1 and whichever of those is left, both being incorrect. Swapping will give you sheep in either case (each possibility with a 1/6th chance, for a 1/3rd chance total)
Choose door 2:
Monty reveals door 3 as incorrect. You WILL get the car if you swap for Door 1. (1/3rd chance).
Choose Door 3:
Monty reveals door 2 as incorrect. You WILL get the car if you swap for Door 1. (1/3rd chance)
If you choose to always stay, you will only have a 1/3rd chance of a car.
How is this related to your question?
If you look at all six probabilities for staying and swapping at once, you will get the answer you saw, and you will see a 1/2 chance of getting the car (2/6 if you do swap plus 1/6 chance if you don't swap). That doesn't show the impact of making a choice.
Using your chart, just compare the red "swap" probabilities to each other to see the impact of always swapping. If you ALWAYS choose to swap, you have a 2/3rd chance of being correct. If you ALWAYS choose to stay, you have a 1/3rd chance to be correct. The possible choices have to be examined independently to assess individual merit of a given choice.
Your question states as if you always choose to swap, but your examples show that you are accounting for both swapping and staying. Once you remove the chance of staying, you are only left with three possible results, not 6.
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4 Answers
4
active
oldest
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4 Answers
4
active
oldest
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active
oldest
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active
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It's simply because your diagram assumes you stay with probability $1/2$. In that case the probability of switching and getting the car is indeed $2/6$, and the probability of switching and not getting the car is $1/6$.
But if you always switch, you're twice as likely to get each of these outcomes - instead of a $1/2$ probability of switching, you have probability $1$ of switching - so they become $2/3$ and $1/3$ respectively.
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add a comment |
$begingroup$
It's simply because your diagram assumes you stay with probability $1/2$. In that case the probability of switching and getting the car is indeed $2/6$, and the probability of switching and not getting the car is $1/6$.
But if you always switch, you're twice as likely to get each of these outcomes - instead of a $1/2$ probability of switching, you have probability $1$ of switching - so they become $2/3$ and $1/3$ respectively.
$endgroup$
add a comment |
$begingroup$
It's simply because your diagram assumes you stay with probability $1/2$. In that case the probability of switching and getting the car is indeed $2/6$, and the probability of switching and not getting the car is $1/6$.
But if you always switch, you're twice as likely to get each of these outcomes - instead of a $1/2$ probability of switching, you have probability $1$ of switching - so they become $2/3$ and $1/3$ respectively.
$endgroup$
It's simply because your diagram assumes you stay with probability $1/2$. In that case the probability of switching and getting the car is indeed $2/6$, and the probability of switching and not getting the car is $1/6$.
But if you always switch, you're twice as likely to get each of these outcomes - instead of a $1/2$ probability of switching, you have probability $1$ of switching - so they become $2/3$ and $1/3$ respectively.
answered Aug 7 '17 at 13:42
Especially LimeEspecially Lime
22.3k22858
22.3k22858
add a comment |
add a comment |
$begingroup$
That indicates that instead of using a strategy, you are randomly choosing whether to swap or stay. The probability of winning a car by doing that is and unsurprising $1/2$.
$$tfrac 13tfrac 12+tfrac 23tfrac 12=tfrac 12$$
The point of the scenario is that if you use strategy of always swapping, the probability of winning a car becomes: $2/3$.
$$tfrac 13tfrac 01+tfrac 23tfrac 11=tfrac 23$$
$endgroup$
2
$begingroup$
The point is that if you switch with probability $1/2$ you become equally likely to pick either of the doors the host hasn't opened - and exactly one of them has a car behind it.
$endgroup$
– Especially Lime
Aug 7 '17 at 13:45
$begingroup$
Indeed...I wrote too soon. :(
$endgroup$
– John Hughes
Aug 7 '17 at 13:45
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Thank you, i get the "always swapping strategy" now but it don't quite understand the sums with two fractions next to each other and how in the top one the probability of getting a car is 1/2, could you explain this please? :)
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– yt.
Aug 7 '17 at 14:01
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It comes dirrect from your diagram, @Ruby. There are two paths on it that lead to winning a car: pick a car ($tfrac 13$) and stay ($tfrac 12$) or pick a sheep $(tfrac 23)$ and swap ($tfrac 12$).
$endgroup$
– Graham Kemp
Aug 7 '17 at 23:30
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@GrahamKemp Okay I see now, and then do you times the two fractions that are next to each other together, and the sum of them is 1/2? If so is it usually written like that or is there a multiplication symbol between the two fractions? Thanks
$endgroup$
– yt.
Aug 8 '17 at 17:56
add a comment |
$begingroup$
That indicates that instead of using a strategy, you are randomly choosing whether to swap or stay. The probability of winning a car by doing that is and unsurprising $1/2$.
$$tfrac 13tfrac 12+tfrac 23tfrac 12=tfrac 12$$
The point of the scenario is that if you use strategy of always swapping, the probability of winning a car becomes: $2/3$.
$$tfrac 13tfrac 01+tfrac 23tfrac 11=tfrac 23$$
$endgroup$
2
$begingroup$
The point is that if you switch with probability $1/2$ you become equally likely to pick either of the doors the host hasn't opened - and exactly one of them has a car behind it.
$endgroup$
– Especially Lime
Aug 7 '17 at 13:45
$begingroup$
Indeed...I wrote too soon. :(
$endgroup$
– John Hughes
Aug 7 '17 at 13:45
$begingroup$
Thank you, i get the "always swapping strategy" now but it don't quite understand the sums with two fractions next to each other and how in the top one the probability of getting a car is 1/2, could you explain this please? :)
$endgroup$
– yt.
Aug 7 '17 at 14:01
$begingroup$
It comes dirrect from your diagram, @Ruby. There are two paths on it that lead to winning a car: pick a car ($tfrac 13$) and stay ($tfrac 12$) or pick a sheep $(tfrac 23)$ and swap ($tfrac 12$).
$endgroup$
– Graham Kemp
Aug 7 '17 at 23:30
$begingroup$
@GrahamKemp Okay I see now, and then do you times the two fractions that are next to each other together, and the sum of them is 1/2? If so is it usually written like that or is there a multiplication symbol between the two fractions? Thanks
$endgroup$
– yt.
Aug 8 '17 at 17:56
add a comment |
$begingroup$
That indicates that instead of using a strategy, you are randomly choosing whether to swap or stay. The probability of winning a car by doing that is and unsurprising $1/2$.
$$tfrac 13tfrac 12+tfrac 23tfrac 12=tfrac 12$$
The point of the scenario is that if you use strategy of always swapping, the probability of winning a car becomes: $2/3$.
$$tfrac 13tfrac 01+tfrac 23tfrac 11=tfrac 23$$
$endgroup$
That indicates that instead of using a strategy, you are randomly choosing whether to swap or stay. The probability of winning a car by doing that is and unsurprising $1/2$.
$$tfrac 13tfrac 12+tfrac 23tfrac 12=tfrac 12$$
The point of the scenario is that if you use strategy of always swapping, the probability of winning a car becomes: $2/3$.
$$tfrac 13tfrac 01+tfrac 23tfrac 11=tfrac 23$$
edited Aug 7 '17 at 13:46
answered Aug 7 '17 at 13:40
Graham KempGraham Kemp
86.4k43479
86.4k43479
2
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The point is that if you switch with probability $1/2$ you become equally likely to pick either of the doors the host hasn't opened - and exactly one of them has a car behind it.
$endgroup$
– Especially Lime
Aug 7 '17 at 13:45
$begingroup$
Indeed...I wrote too soon. :(
$endgroup$
– John Hughes
Aug 7 '17 at 13:45
$begingroup$
Thank you, i get the "always swapping strategy" now but it don't quite understand the sums with two fractions next to each other and how in the top one the probability of getting a car is 1/2, could you explain this please? :)
$endgroup$
– yt.
Aug 7 '17 at 14:01
$begingroup$
It comes dirrect from your diagram, @Ruby. There are two paths on it that lead to winning a car: pick a car ($tfrac 13$) and stay ($tfrac 12$) or pick a sheep $(tfrac 23)$ and swap ($tfrac 12$).
$endgroup$
– Graham Kemp
Aug 7 '17 at 23:30
$begingroup$
@GrahamKemp Okay I see now, and then do you times the two fractions that are next to each other together, and the sum of them is 1/2? If so is it usually written like that or is there a multiplication symbol between the two fractions? Thanks
$endgroup$
– yt.
Aug 8 '17 at 17:56
add a comment |
2
$begingroup$
The point is that if you switch with probability $1/2$ you become equally likely to pick either of the doors the host hasn't opened - and exactly one of them has a car behind it.
$endgroup$
– Especially Lime
Aug 7 '17 at 13:45
$begingroup$
Indeed...I wrote too soon. :(
$endgroup$
– John Hughes
Aug 7 '17 at 13:45
$begingroup$
Thank you, i get the "always swapping strategy" now but it don't quite understand the sums with two fractions next to each other and how in the top one the probability of getting a car is 1/2, could you explain this please? :)
$endgroup$
– yt.
Aug 7 '17 at 14:01
$begingroup$
It comes dirrect from your diagram, @Ruby. There are two paths on it that lead to winning a car: pick a car ($tfrac 13$) and stay ($tfrac 12$) or pick a sheep $(tfrac 23)$ and swap ($tfrac 12$).
$endgroup$
– Graham Kemp
Aug 7 '17 at 23:30
$begingroup$
@GrahamKemp Okay I see now, and then do you times the two fractions that are next to each other together, and the sum of them is 1/2? If so is it usually written like that or is there a multiplication symbol between the two fractions? Thanks
$endgroup$
– yt.
Aug 8 '17 at 17:56
2
2
$begingroup$
The point is that if you switch with probability $1/2$ you become equally likely to pick either of the doors the host hasn't opened - and exactly one of them has a car behind it.
$endgroup$
– Especially Lime
Aug 7 '17 at 13:45
$begingroup$
The point is that if you switch with probability $1/2$ you become equally likely to pick either of the doors the host hasn't opened - and exactly one of them has a car behind it.
$endgroup$
– Especially Lime
Aug 7 '17 at 13:45
$begingroup$
Indeed...I wrote too soon. :(
$endgroup$
– John Hughes
Aug 7 '17 at 13:45
$begingroup$
Indeed...I wrote too soon. :(
$endgroup$
– John Hughes
Aug 7 '17 at 13:45
$begingroup$
Thank you, i get the "always swapping strategy" now but it don't quite understand the sums with two fractions next to each other and how in the top one the probability of getting a car is 1/2, could you explain this please? :)
$endgroup$
– yt.
Aug 7 '17 at 14:01
$begingroup$
Thank you, i get the "always swapping strategy" now but it don't quite understand the sums with two fractions next to each other and how in the top one the probability of getting a car is 1/2, could you explain this please? :)
$endgroup$
– yt.
Aug 7 '17 at 14:01
$begingroup$
It comes dirrect from your diagram, @Ruby. There are two paths on it that lead to winning a car: pick a car ($tfrac 13$) and stay ($tfrac 12$) or pick a sheep $(tfrac 23)$ and swap ($tfrac 12$).
$endgroup$
– Graham Kemp
Aug 7 '17 at 23:30
$begingroup$
It comes dirrect from your diagram, @Ruby. There are two paths on it that lead to winning a car: pick a car ($tfrac 13$) and stay ($tfrac 12$) or pick a sheep $(tfrac 23)$ and swap ($tfrac 12$).
$endgroup$
– Graham Kemp
Aug 7 '17 at 23:30
$begingroup$
@GrahamKemp Okay I see now, and then do you times the two fractions that are next to each other together, and the sum of them is 1/2? If so is it usually written like that or is there a multiplication symbol between the two fractions? Thanks
$endgroup$
– yt.
Aug 8 '17 at 17:56
$begingroup$
@GrahamKemp Okay I see now, and then do you times the two fractions that are next to each other together, and the sum of them is 1/2? If so is it usually written like that or is there a multiplication symbol between the two fractions? Thanks
$endgroup$
– yt.
Aug 8 '17 at 17:56
add a comment |
$begingroup$
You're calculating the probability wrong. If you decide at random with equal probability to swap or stay, in $tfrac{2}{6}$ of all cases you will have swapped and gotten the car.
But that is not the probability of if you decide to swap, would you get the car?
Note that in exactly $tfrac{3}{6}=tfrac{1}{2}$ of all cases, you will have swapped, and in $tfrac{1}{2}$ of all cases you will have stayed. But look at those cases.
Of the $tfrac{3}{6}$ cases where you will have swapped, there is a $tfrac{2}{6}$ chance of having swapped and gotten a car while there is a $tfrac{1}{6}$ of having swapped and gotten a goat sheep.
On the other hand there is a $tfrac{1}{6}$ chance of having stayed and gotten a car while there is a $tfrac{2}{6}$ of having swapped and gotten a sheep.
So if you swap, the probability of getting a car is twice that of getting a sheep, while if you stay, the chance of getting a sheep is twice that of getting a car.
Which is the exact outcome of the canonical Monty Hall problem.
$endgroup$
add a comment |
$begingroup$
You're calculating the probability wrong. If you decide at random with equal probability to swap or stay, in $tfrac{2}{6}$ of all cases you will have swapped and gotten the car.
But that is not the probability of if you decide to swap, would you get the car?
Note that in exactly $tfrac{3}{6}=tfrac{1}{2}$ of all cases, you will have swapped, and in $tfrac{1}{2}$ of all cases you will have stayed. But look at those cases.
Of the $tfrac{3}{6}$ cases where you will have swapped, there is a $tfrac{2}{6}$ chance of having swapped and gotten a car while there is a $tfrac{1}{6}$ of having swapped and gotten a goat sheep.
On the other hand there is a $tfrac{1}{6}$ chance of having stayed and gotten a car while there is a $tfrac{2}{6}$ of having swapped and gotten a sheep.
So if you swap, the probability of getting a car is twice that of getting a sheep, while if you stay, the chance of getting a sheep is twice that of getting a car.
Which is the exact outcome of the canonical Monty Hall problem.
$endgroup$
add a comment |
$begingroup$
You're calculating the probability wrong. If you decide at random with equal probability to swap or stay, in $tfrac{2}{6}$ of all cases you will have swapped and gotten the car.
But that is not the probability of if you decide to swap, would you get the car?
Note that in exactly $tfrac{3}{6}=tfrac{1}{2}$ of all cases, you will have swapped, and in $tfrac{1}{2}$ of all cases you will have stayed. But look at those cases.
Of the $tfrac{3}{6}$ cases where you will have swapped, there is a $tfrac{2}{6}$ chance of having swapped and gotten a car while there is a $tfrac{1}{6}$ of having swapped and gotten a goat sheep.
On the other hand there is a $tfrac{1}{6}$ chance of having stayed and gotten a car while there is a $tfrac{2}{6}$ of having swapped and gotten a sheep.
So if you swap, the probability of getting a car is twice that of getting a sheep, while if you stay, the chance of getting a sheep is twice that of getting a car.
Which is the exact outcome of the canonical Monty Hall problem.
$endgroup$
You're calculating the probability wrong. If you decide at random with equal probability to swap or stay, in $tfrac{2}{6}$ of all cases you will have swapped and gotten the car.
But that is not the probability of if you decide to swap, would you get the car?
Note that in exactly $tfrac{3}{6}=tfrac{1}{2}$ of all cases, you will have swapped, and in $tfrac{1}{2}$ of all cases you will have stayed. But look at those cases.
Of the $tfrac{3}{6}$ cases where you will have swapped, there is a $tfrac{2}{6}$ chance of having swapped and gotten a car while there is a $tfrac{1}{6}$ of having swapped and gotten a goat sheep.
On the other hand there is a $tfrac{1}{6}$ chance of having stayed and gotten a car while there is a $tfrac{2}{6}$ of having swapped and gotten a sheep.
So if you swap, the probability of getting a car is twice that of getting a sheep, while if you stay, the chance of getting a sheep is twice that of getting a car.
Which is the exact outcome of the canonical Monty Hall problem.
edited Aug 8 '17 at 7:24
answered Aug 7 '17 at 17:27
SQBSQB
1,74611026
1,74611026
add a comment |
add a comment |
$begingroup$
The key to the solution of the Monty Hall problem is that Monty Hall reveals one of the incorrect solutions after stage one. The problem ends up being exclusively about the door that you choose versus the one that is left.
This means that the probabilities are as follows:
We will assume Door 1 is the car and Door 2 and 3 are sheep.
Choose Door 1:
Monty can reveal door 2 or 3, so you have a choice between 1 and whichever of those is left, both being incorrect. Swapping will give you sheep in either case (each possibility with a 1/6th chance, for a 1/3rd chance total)
Choose door 2:
Monty reveals door 3 as incorrect. You WILL get the car if you swap for Door 1. (1/3rd chance).
Choose Door 3:
Monty reveals door 2 as incorrect. You WILL get the car if you swap for Door 1. (1/3rd chance)
If you choose to always stay, you will only have a 1/3rd chance of a car.
How is this related to your question?
If you look at all six probabilities for staying and swapping at once, you will get the answer you saw, and you will see a 1/2 chance of getting the car (2/6 if you do swap plus 1/6 chance if you don't swap). That doesn't show the impact of making a choice.
Using your chart, just compare the red "swap" probabilities to each other to see the impact of always swapping. If you ALWAYS choose to swap, you have a 2/3rd chance of being correct. If you ALWAYS choose to stay, you have a 1/3rd chance to be correct. The possible choices have to be examined independently to assess individual merit of a given choice.
Your question states as if you always choose to swap, but your examples show that you are accounting for both swapping and staying. Once you remove the chance of staying, you are only left with three possible results, not 6.
$endgroup$
add a comment |
$begingroup$
The key to the solution of the Monty Hall problem is that Monty Hall reveals one of the incorrect solutions after stage one. The problem ends up being exclusively about the door that you choose versus the one that is left.
This means that the probabilities are as follows:
We will assume Door 1 is the car and Door 2 and 3 are sheep.
Choose Door 1:
Monty can reveal door 2 or 3, so you have a choice between 1 and whichever of those is left, both being incorrect. Swapping will give you sheep in either case (each possibility with a 1/6th chance, for a 1/3rd chance total)
Choose door 2:
Monty reveals door 3 as incorrect. You WILL get the car if you swap for Door 1. (1/3rd chance).
Choose Door 3:
Monty reveals door 2 as incorrect. You WILL get the car if you swap for Door 1. (1/3rd chance)
If you choose to always stay, you will only have a 1/3rd chance of a car.
How is this related to your question?
If you look at all six probabilities for staying and swapping at once, you will get the answer you saw, and you will see a 1/2 chance of getting the car (2/6 if you do swap plus 1/6 chance if you don't swap). That doesn't show the impact of making a choice.
Using your chart, just compare the red "swap" probabilities to each other to see the impact of always swapping. If you ALWAYS choose to swap, you have a 2/3rd chance of being correct. If you ALWAYS choose to stay, you have a 1/3rd chance to be correct. The possible choices have to be examined independently to assess individual merit of a given choice.
Your question states as if you always choose to swap, but your examples show that you are accounting for both swapping and staying. Once you remove the chance of staying, you are only left with three possible results, not 6.
$endgroup$
add a comment |
$begingroup$
The key to the solution of the Monty Hall problem is that Monty Hall reveals one of the incorrect solutions after stage one. The problem ends up being exclusively about the door that you choose versus the one that is left.
This means that the probabilities are as follows:
We will assume Door 1 is the car and Door 2 and 3 are sheep.
Choose Door 1:
Monty can reveal door 2 or 3, so you have a choice between 1 and whichever of those is left, both being incorrect. Swapping will give you sheep in either case (each possibility with a 1/6th chance, for a 1/3rd chance total)
Choose door 2:
Monty reveals door 3 as incorrect. You WILL get the car if you swap for Door 1. (1/3rd chance).
Choose Door 3:
Monty reveals door 2 as incorrect. You WILL get the car if you swap for Door 1. (1/3rd chance)
If you choose to always stay, you will only have a 1/3rd chance of a car.
How is this related to your question?
If you look at all six probabilities for staying and swapping at once, you will get the answer you saw, and you will see a 1/2 chance of getting the car (2/6 if you do swap plus 1/6 chance if you don't swap). That doesn't show the impact of making a choice.
Using your chart, just compare the red "swap" probabilities to each other to see the impact of always swapping. If you ALWAYS choose to swap, you have a 2/3rd chance of being correct. If you ALWAYS choose to stay, you have a 1/3rd chance to be correct. The possible choices have to be examined independently to assess individual merit of a given choice.
Your question states as if you always choose to swap, but your examples show that you are accounting for both swapping and staying. Once you remove the chance of staying, you are only left with three possible results, not 6.
$endgroup$
The key to the solution of the Monty Hall problem is that Monty Hall reveals one of the incorrect solutions after stage one. The problem ends up being exclusively about the door that you choose versus the one that is left.
This means that the probabilities are as follows:
We will assume Door 1 is the car and Door 2 and 3 are sheep.
Choose Door 1:
Monty can reveal door 2 or 3, so you have a choice between 1 and whichever of those is left, both being incorrect. Swapping will give you sheep in either case (each possibility with a 1/6th chance, for a 1/3rd chance total)
Choose door 2:
Monty reveals door 3 as incorrect. You WILL get the car if you swap for Door 1. (1/3rd chance).
Choose Door 3:
Monty reveals door 2 as incorrect. You WILL get the car if you swap for Door 1. (1/3rd chance)
If you choose to always stay, you will only have a 1/3rd chance of a car.
How is this related to your question?
If you look at all six probabilities for staying and swapping at once, you will get the answer you saw, and you will see a 1/2 chance of getting the car (2/6 if you do swap plus 1/6 chance if you don't swap). That doesn't show the impact of making a choice.
Using your chart, just compare the red "swap" probabilities to each other to see the impact of always swapping. If you ALWAYS choose to swap, you have a 2/3rd chance of being correct. If you ALWAYS choose to stay, you have a 1/3rd chance to be correct. The possible choices have to be examined independently to assess individual merit of a given choice.
Your question states as if you always choose to swap, but your examples show that you are accounting for both swapping and staying. Once you remove the chance of staying, you are only left with three possible results, not 6.
edited Aug 8 '17 at 17:20
answered Aug 7 '17 at 19:07
AvioseAviose
1513
1513
add a comment |
add a comment |
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$begingroup$
"In my tree diagram i haven't included the host revealing a door, so could this be a factor as to why my answer is wrong?" Yes.
$endgroup$
– John Hughes
Aug 7 '17 at 13:40
$begingroup$
What happens on the other 3/6?
$endgroup$
– Kevin
Aug 8 '17 at 13:07
$begingroup$
@JohnHughes Actually, the host revealing a door is taken into account with swapping-toggles-the-cariness property of the tree diagram.
$endgroup$
– wizzwizz4
Aug 8 '17 at 15:41
$begingroup$
@Kevin the other 3/6 is the probability of getting sheep
$endgroup$
– yt.
Aug 8 '17 at 17:35