/sci/ - Science & Math

N mod 3..n = 2/3

So long as the decision set initially has three vectors and you get to scry 1 (MTG Ruleset), opting to change will always yield 1/3 + 1/3 = 2/3

>>10836784
watch the newest vsauce (d!ng) video about it if you haven't already, its the clearest explanation i've seen

>>10836794
>>10836790
No, I'm aware that this is how it works and I agree that this is the correct total probability. What I don't understand is why we use total probability to determine the probability of winning on either staying and switching when it comes down to that, since Monty will always remove a goat from the equation anyway. It seems very misleading to consider the third door relevant to the choice of staying or switching.


For example;

A -> Goat
B -> Car
B -> Goat
A -> Car

Staying or switching is no more than selecting the remaining A or B, since C (or any goat option that monty takes out) is something that was never a choice. It always comes down to deciding between a Goat or a Car.

>>10836842
>What I don't understand is why we use total probability to determine the probability of winning
What do you mean by this? What do you think the "total probability" represents if not the probability of winning?

>>10836842
>It seems very misleading to consider the third door relevant to the choice of staying or switching.
It's extremely relevant because your initial choice is actually a fair, random option of 1 out of 3 equal possibilities.
The remaining door you didn't pick that the host didn't open is NOT a fair, random option of 1 out of 2 equal possibilities because the host will NEVER show you a winning door with a car behind it.
3 possibilities here:
1) Your initial pick was the winner
2) Your initial pick was the first loser door
3) Your initial pick was the second loser door
If 2 OR 3 is true, the host MUST show you the remaining loser door because the other one had the car and it would ruin the game to tell you where the car is.
If 1 is true then the host actually had a choice of showing you either of the two loser doors you didn't initially pick.
If 2 is true and you switch then you win.
If 3 is true and you switch then you win.
If 1 is true and you switch then you lose.

>>10836864
The probability after being padded out with the third door.

For example, if I'm presented with 100 people and I am told to find the single criminal between amongst them, but before I can make any relevant decision on picking someone, they start to remove 98 innocent people, leaving me with one innocent person and one guilty person. Why would the other 98 people be taken into account for the probability of whenever I get the right person or not, especially if my selection is every time trimmed down one innocent and one guilty person? Doesn't that just mean my chance to select the right person is 1/2 every time?

>>10836842
>Monty will always remove a goat from the equation anyway.
Yeah but which goat? Most of the time he doesn't get to pick, because you left him with only one remaining. So by opening the goat door he told you something about the remaining door -- but not anything about the door you picked since he couldn't open it even if it had a goat.

Another way of looking at it is to ignore the door opening entirely.
What if you picked a door and then Monty, without doing anything else, offered you the option to either open your door or open BOTH remaining doors?
This is essentially the same exact choice as the original Monty Hall problem, but in this formulation it should be intuitively obvious why switching gives you a 66% probability. Two doors are better than one. Whether Monty opens a goat door among those two beforehand doesn't make that probability any smaller, because you already knew at least one must have been a goat.

>>10836892
>but before I can make any relevant decision on picking someone, they start to remove 98 innocent people, leaving me with one innocent person and one guilty person.
Sure, in that case it's a 50% chance. But it's not the same problem.
If instead before removing 98 innocent people they rolled a 100 sided die to select one person to be exempt from the removal process regardless of guilt or innocence, now you've got a 99% chance that the one other guy they didn't remove is the guilty one, because 99% of the time they have no choice whatsoever in which 98 people to remove; one is guilty and one is exempt.

>>10836892
>For example, if I'm presented with 100 people and I am told to find the single criminal between amongst them, but before I can make any relevant decision on picking someone, they start to remove 98 innocent people
There's your problem. That is NOT how it works.
You get TWO picks.
You would first pick 1 out of 100 at random.
THEN they would do the 98 door elimination.
Now if you know YOUR first pick is genuinely a random 1/100 choice while THEIR door elimination would ONLY leave behind a losing door in the case where you got it right on your first 1/100 pick, then of course you should switch and make your second and final choice the door neither you initially picked nor did they reveal as a loser door. There would be a 1/100 chance your initial door was right and a 99/100 chance the remaining door they didn't reveal to you is right.

>>10836842
Because every binary branch will always end in a vertex graph output.

>

>>10836784
50/50
50% ALWAYS

>>10836784
Whether it's 1/2 or 1/3 depends on whether the person that opens the door is omniscient and intentionally opens a door with a goat, or is not omniscient and opens a door without knowing what is behind it and that door then happens to have a goat.

>>10837085
t. brainlet
>>10837113
You don't need to be omniscient to know where the prize is on your own show.
If you guys want to do a proper nitpicking here there are a couple of minor issues with the Monty Hall problem:
A) Despite the name that's not actually how the Let's Make a Deal Show actually worked. Not an issue as long as you don't try to use the scenarios from that show as your basis for answering how this hypothetical Monty Hall problem would play out.
B) You must make the assumption that every single round the host always opens a door and offers to let you switch. If you don't make that assumption then the game is completely different since the host could just let you instantly lose after your initial pick every time your initial pick is a goat, and then staying rather than switching might be the right answer as high as 100% of the time if the host only ever gives you the option to the switch when he isn't able to insta-kill your chances over your first pick for being a goat.
As long as you take those two issues into account though then it's definitely a 2/3 vs. 1/3 advantage for switching every time. The host knows what's behind all three doors, he's always forced to open one of the remaining doors after your initial pick, and he will never open the door with the car when opening one of the remaining doors after your initial pick because doing so would defeat the purpose of even asking if you want to switch.
All this being true you either:
1) Picked the winner first.
2) Picked loser A first.
3) Picked loser B first.
If 1 is true, host can reveal either loser A or loser B, and switching will result in a loss.
If 2 is true, host must reveal loser B and switching will result in a win.
If 3 is true, host must reveal loser A and switching will result in a win.
2/3 possibilities result in a win if you switch. 1/3 possibilities result in a loss if you switch.

>>10836784
Imagine its 1000 doors instead 3.

>>10836842
>It seems very misleading to consider the third door relevant to the choice of staying or switching.
Concept is misleading on purpose by design and its working. It make third door appear irrelevant but you should figure out it is relevant unless you are brainlet.

>>10836892
You actually modified scenario to make it easier to get and you just didn't. You are hopeless.

easiest way to think about it is:

you pick a door from the three at random. you have a 33% chance of win. the other two doors each have a 33% chance of win, for 66% combined

Monty opens one of the other two doors, revealing a goat. that door's chance is now zero. but yours hasn't changed, yours is still 33%. so therefore the remaining door must be 67%.

>>10836871
This is probably the best explanation of this that I have ever read. Thanks anon

>yfw you realize this is an aeonic question meant to help you overthrow the tyranny of number

In 9th grade my teacher brought this up to us I sat in the back of the class when everyone raise their hand but me saying they understood it I realized that I was being executed

>>10836784

Here is your misunderstanding.

You ARE correct about a 50-50 chance being behind either door *IF* you did NOT know about the previous existing conditions (3 doors, goats behind 2, door with goat shown).

Statistics is about reasoning with *ALL* the knowledge you have. When you include the extra knowledge you have THEN the choice becomes obvious.

>>10836784
Dude just look at it this way, the first time you pick you either pick a goat door or a car door. If you pick a goat door you always win if you switch since Monty Hall will always open the other goat door. If you pick a car door first time around you always lose if you switch. Hence the chance of winning if you always switch is the chance of picking a goat door first time around, which is 2/3.

>>10836784
50%

>>10836892
>if I'm presented with 100 people and I am told to find the single criminal between amongst them, but before I can make any relevant decision on picking someone, they start to remove 98 innocent people, leaving me with one innocent person and one guilty person.
This is not Monty Hall Problem scenario. You make choice first and then you are offered with possibility to make correction. If you can't notice diffrence you will never figure out why switching in Monty Hall is better option.

>>10836784
Imagine 500000 doors in front of you and then throw a stone between them, and which ever one it hits is your choice. Then somebody comes over and tell you ok it's either the door the stone hit or the one I exclaim next. What would you choose?

>>10836784
50/50, it either is or is not

>>10836784
Wikipedia has a page on the monty hall problem with multiple visual aides. I think this is my favorite.
(Context, this visual aide assumes the player picks door #1 the first time)

If those don't satisfy you, then https://www.youtube.com/watch?v=6wS5xOZ7Rq8

What if we erase the memory of their being three doors from the contestant? So the goat is showing and they pick either door 2 or 3. What are the odds then? Host still has memories.

>>10842773
The odds are 2/3 that the contestant's first pick was the right one.
The odds are 1/2 that the contestant will pick the right one when in a condition where they no longer know which of those two remaining doors was the one they initially picked.

>>10842773
The odds are 1/3 that the contestant's first pick was the right one.
The odds are 2/3 that the contestant will when if their second pick is the remaining door they didn't initially pick.
The odds are 1/2 that the contestant will pick the right one when in a condition where they no longer know which of those two remaining doors was the one they initially picked.

>>10842773
>>10842780
The odds are 2/3 that the contestant will *win* if their second pick is the remaining door they didn't initially pick.

>>10841709
autism b gone

Wasn't the real Monty Hall problem that the results on the show didn't match the statistics, so it was obvious that Monty was cheating? Perhaps something not unlike my 99.9% strikeout rate?

>>10842795
The show didn't do what the problem does. Totally different.

>>10842795
>Wasn't the real Monty Hall problem that the results on the show didn't match the statistics, so it was obvious that Monty was cheating?
No, where did you even get that bizarre idea from?
The Monty Hall Problem had very little to do with the actual Let's Make a Deal show. He didn't even offer to switch doors on the show. His offer was giving you cash as an alternative to getting whatever will end up revealed behind the door you picked.

>>10842843

No, when you pick from 3 closed doors, 2 worthless, 1 with a prize. You have a greater chance on the first pick that it something worthless. So you always should pick from the remaining 2 if given the chance.

>>10842857
What are you talking about? I'm just saying the show Let's Make a Deal wasn't the same as the Monty Hall Problem.

>>10842857
You're describing the Monty Hall problem. The anon you replied to was describing the actual show and how it didn't have the same premise as the Monty Hall problem.

>>10836784
Conditioning is bullshit. You create a fake event with words in your mind and assign it a number ("conditional probability") which means the intensity of the "magical force" that "push the event into being true".
Throw away this garbage and think in concrete terms and you'll solve every of these problems.

"what happens if I do it bazillion times" is how to think about it.

If you play the monty hall game 9 million times and get the first choice wrong between 5990000 and 6010000 times, then by switching you get away with the prize between 5990000 and 6010000 times. All logic and basic arithmetics. No probabilities involved.

Now if you want to have probabilities involved, let X_1,...,X_N be independent random variables taking values in {0,1}, and p a real number such that for every k, P(X_k=1)=p.From Hoeffding's inequality we get that for every r>0, P(|S_N-Np|> r) <= 2*exp (-2r^2/N).
Setting p=2/3, r = 10000 and N=9000000 gives a bound =~ 4.4*10^{-10}. Said otherwise, you can treat the number of the story above as granted by nature.

There is win.

You pick up a door.

There is 33% probability...

One door is opened, there is goat.

There is a 50% probability that there is a win in a door, any after it.

Just television ruined your lives dumbfucks.

What if you want the goat?

>>10836784
It's easiest if you make the same experiment with 100 doors. Then the intuition kicks in right away.

1) start with 100 doors, you can pick one of them
2) the host opens the other 98 doors
3) what's the chance that the only other door the host hasn't opened is the correct one, compared to the chance that you picked the correct one?

Your door has 1/100 chance of success while if you switch you have 99/100 chance of success. So you can easily see how switching is preferable each and every time.

>>10842881
>guys i need help doing simple probabilities
>oh thats super easy just do this unnecessary more complex way of solving it
/sci/ in one post. Super autists who are smart but too stupid to explain simple shit to anyone

>>10836784
If you start off with 1 million doors and all but 1 is a goat, you pick a door, and then Monty hall opens every doorexcept the one you picked and one other and tells you the winner is behind one of these two doors and you have the opportunity to switch, would you?
It still boils down to 2 doors just like you said.
But your original choice was a 1 in a million chance. Do you stick with your original choice?
Of course not

>>10843693
If it was simple, you wouldn't have had so many people with math degree fail on this problem when it was made public for the first time.
The second part of the post you're referring is just a technical justification of the law of large numbers used in the statement.
If tldr, you just say that "after 9 000 000 trials,the player fails to spot the prize about 6 000 000 times" and the rest of the reasoning holds.

>>10843387

>What if you want the goat?