Awesome Pictures Thread

[apocrypha]

I love these.

[Rishathra]

The guy who makes those does some pretty cool stuff.  I like to check in every few weeks to see what else he's come up with.

[Raging Turtle]

Not going to read it.  That's a case of math missing the point.  Regardless of what you do, you still only get one door out of three.

Many readers refused to believe that switching is beneficial. After the Monty Hall problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine claiming that vos Savant was wrong. (Tierney 1991) Even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy

 

Read the article and do the card trick it suggests as proof.  It's not intuitive, but it's also not complicated.

[Morat20]

Heck, you're here. Can you write code? Simulate the problem a million times, switching half the time and staying half the time.

Compare the results.

It's not hard to simulate, and the numbes bear it out -- switching wins 2/3rds of the time under the classic setup. (Which was never, IIRC, actually ever used on the game show). Staying wins 1/3rd of the time. Seriously, you can bang it out in C in like fifteen minutes. Five of which will be "how the fuck does the goddamn random() shit work again?"

There's also the million-door varient, which (to me) makes it ridiculously clear. You have a million doors. Behind one is a prize. You pick a door. The host throws open all but one of the remaining doors, showing nada. Should you switch or should you stay? You should switch.

Also, in general -- if it's a choice between "what the math says" and "what my gut says" on a matter of basic probability, go with the math. Vegas makes tons of money off people who go with their guts.

[Miasma]

Not going to read it.  That's a case of math missing the point.  Regardless of what you do, you still only get one door out of three.

No, you get one choice of three doors and then a second choice of two doors.  I'm going to assume you are trolling by sticking your head in the sand and refusing to believe the truth on the very scenario designed to do exactly that.

[Mosesandstick]

I think drawing a probability tree is one of the easier ways of being sure about the switch.

[Morfiend]

There's also the million-door varient, which (to me) makes it ridiculously clear. You have a million doors. Behind one is a prize. You pick a door. The host throws open all but one of the remaining doors, showing nada. Should you switch or should you stay? You should switch.

When I first heard this problem I thought as Cyrrex does now, then someone explained the million door version and I had that "aha" or lightbulb moment. It comes down to 1 of 3 ONLY if you dont change doors.

You should switch

[Morat20]

There's also the million-door varient, which (to me) makes it ridiculously clear. You have a million doors. Behind one is a prize. You pick a door. The host throws open all but one of the remaining doors, showing nada. Should you switch or should you stay? You should switch.

When I first heard this problem I thought as Cyrrex does now, then someone explained the million door version and I had that "aha" or lightbulb moment. It comes down to 1 of 3 ONLY if you dont change doors.

You should switch

Effectively, throwing open the doors is a smokescreen -- it's designed to confuse people. (In fact, it's one of a handful of problems designed to teach students that their instincts on probabilty are often wrong). The real choice is "Do you want the original door you picked, or all the other doors combined? The prize never moved".

You really want "all the other doors combined".

[Sky]

On the monty haul thing - I didn't realize one of the remaining doors was opened to show no prize. In that case, it's obvious you switch. I thought it was based on the doors remaining closed.

[Lantyssa]

[Engels]

About the 1 million door illustration that clarified my 'intuitive issue':

From the wiki

This [million door] example can also be used to illustrate the opposite situation in which the host does not know where the prize is and opens doors randomly. There is a 999,999/1,000,000 probability that the contestant selects wrong initially, and the prize is behind one of the other doors. If the host goes about randomly opening doors not knowing where the prize is, the probability is likely that the host will reveal the prize before two doors are left (the contestant's choice and one other) to switch between. If the host happens to not reveal the car, then both of the remaining doors have an equal probability of containing a car.

Emphasis mine. In other words, if the host knows where the car is, then the probability increases. If on the other hand, the host doesn't know, then the probability remains the same.

Most people going with the 'gut' instinct assume, even though its explicitly said at the start, that the host is picking randomly, when in fact he's specifically knows to not pick the car door. That's what reduces the probability, not the actual switching itself, which in the 3 door solution seems pure magical thinking, and hence the refusal to 'believe'.

[Morat20]

About the 1 million door illustration that clarified my 'intuitive issue':

From the wiki

This [million door] example can also be used to illustrate the opposite situation in which the host does not know where the prize is and opens doors randomly. There is a 999,999/1,000,000 probability that the contestant selects wrong initially, and the prize is behind one of the other doors. If the host goes about randomly opening doors not knowing where the prize is, the probability is likely that the host will reveal the prize before two doors are left (the contestant's choice and one other) to switch between. If the host happens to not reveal the car, then both of the remaining doors have an equal probability of containing a car.

Emphasis mine. In other words, if the host knows where the car is, then the probability increases. If on the other hand, the host doesn't know, then the probability remains the same.

Most people going with the 'gut' instinct assume, even though its explicitly said at the start, that the host is picking randomly, when in fact he's specifically knows to not pick the car door. That's what reduces the probability, not the actual switching itself, which in the 3 door solution seems pure magical thinking, and hence the refusal to 'believe'.

Huh. I thought it was explicit that the host deliberately revealed non-prize doors.

[Lantyssa]

Emphasis mine. In other words, if the host knows where the car is, then the probability increases. If on the other hand, the host doesn't know, then the probability remains the same.

There is an equal probability only if the host has a chance of opening the door to reveal the car, in which case you've already lost.

The point of the game is that the door opened does not have the car.  Maybe the host doesn't know, but his assistant pops out from the door.  Your odds are still better to switch doors.

Your pick: 1 in 3 chance.
Host pick: 2 in 3 chance.

Since the game eliminates the bad door, you are meant to think, "Oh, it's 50/50!".  However, the door eliminated is dependent upon what you picked.  That's the trick.  Your pick is random.  The elimination is not.  There is still a 2 in 3 chance of the car being behind the host's door because it was part of a larger set.

[Mosesandstick]

Huh. I thought it was explicit that the host deliberately revealed non-prize doors.

Definitely, but it's also usually brought up when taught so that students can think about how it makes the problem different.

[Prospero]

About the 1 million door illustration that clarified my 'intuitive issue':

From the wiki

This [million door] example can also be used to illustrate the opposite situation in which the host does not know where the prize is and opens doors randomly. There is a 999,999/1,000,000 probability that the contestant selects wrong initially, and the prize is behind one of the other doors. If the host goes about randomly opening doors not knowing where the prize is, the probability is likely that the host will reveal the prize before two doors are left (the contestant's choice and one other) to switch between. If the host happens to not reveal the car, then both of the remaining doors have an equal probability of containing a car.

Emphasis mine. In other words, if the host knows where the car is, then the probability increases. If on the other hand, the host doesn't know, then the probability remains the same.

Most people going with the 'gut' instinct assume, even though its explicitly said at the start, that the host is picking randomly, when in fact he's specifically knows to not pick the car door. That's what reduces the probability, not the actual switching itself, which in the 3 door solution seems pure magical thinking, and hence the refusal to 'believe'.

Huh. I thought it was explicit that the host deliberately revealed non-prize doors.

It is. The variant that Engles is using is showing the case where the host does not know which door has the prize and accidentally opens 999,998 doors with goats behind them. Also from the wiki, describing the standard Vos Savant solution.

It may be easier to appreciate the solution by considering the same problem with 1,000,000 doors instead of just three (vos Savant 1990). In this case there are 999,999 doors with goats behind them and one door with a prize. The player picks a door. His initial probability of winning is 1 out of 1,000,000. The game host then opens 999,998 of the other doors revealing 999,998 goats. (Imagine the host starting with the first door and going down a line of 1,000,000 doors, opening each one, skipping over only the player's door and one other door.) The host then offers the player the chance to switch to the only other unopened door. On average, in 999,999 out of 1,000,000 times the other door will contain the prize, as 999,999 out of 1,000,000 times the player first picked a door with a goat.

[Engels]

It is explicit, but it can be overlooked when thinking about the problem 'too mathematically', if you will.

Lets pose the problem entirely differently, eliminating the host altogether to remove the implicit psychological factor that the host wants you to lose:

Assume you are both player and host and you have a million doors. You pick one. Chances are at 1/1M. You then randomly open all remaining doors but one. Statistically, chances of accidentally opening the door with the car are 999,998 out of 999,999. On the rare occurrence that you have NOT picked the car by accident (1/999,999), then the remaining choices are indeed 1/2, which is better than the initial probability of 1/1M. Hence 'switching' is good, but if you take into account that you had to go through billions of times where you randomly accidentally chose the door with the car and revealed it, then the chances are still just as awful as the initial 1/1M.

So, introducing the host into the picture and forcefully eliminating the second random choice of 1/999,999, then you have in fact removed the bad chances. In the million door case, the 1/999,999 probability and in the 3 door case, the 1/3 probability.

[Prospero]

Correct. If you solve a different problem than the Monty Haul problem you shockingly enough get a different answer.

[Engels]

But that's what's psychologically interesting about it. The mathematics behind it aren't that hard, but what is hard is to grasp why people pick the wrong answer. Its not simply the bias towards sticking with your first decision; its obfuscated by misunderstandings regarding the problem's first assumptions.

[Morat20]

When taught in class, the problem is obvious. Three doors, it's 1/3. Two doors, it's 50/50. That's where people's gut understanding of statistics begins and ends.

Most people, unless they've trained up to it, simply can't instinctively handle two or more step statistical problems instinctively. They can handle a simple, straightforward case -- but add in changes, hidden information or partial information (or what's exposed as the problem progresses) and people's instincts fail them.

I guess it's the difference between a unique event and a series of related statistical events.

[Prospero]

Sorry Engles, I mixed you up with Cyrrex there and thought you were arguing that the math was wrong. My bad.

[Engels]

This whole thing should probably be branched into useless conversation or something, cuz there's not a Awesome picture in sight. To that effect:

Awsome 1/2 probability goat:

[01101010]

More Marty please...  oh wait, he is probably more Funny.

[Morat20]

That goat is fairly awesome. It will blend in well with the zebras.

[Sky]

More Marty please... 

[MahrinSkel]

Not going to read it.  That's a case of math missing the point.  Regardless of what you do, you still only get one door out of three.

Here's the key to understanding the Monty Hall problem: There's one parameter that is implied, but never stated: Monty knows which doors have goats.  So when he opens a door with a goat, you think you haven't actually learned anything (because you already knew one of the doors you weren't taking had a goat), but you have (the "not goat" potential of the door left over has increased).  Your instinctive judgement of the chances is based on Monty picking a door at random, but he's not.

--Dave

[Merusk]

And that's the part everyone never understand until it's explained.  As soon as you do that, suddenly people say "oooh."   Everyone makes different assumptions.  I always assumed the scenario was out to screw me and he was just a clueless cog when offering the chance to switch doors, which is why I never understood "you're now picking from 2 doors."

[lamaros]

It still surprises me when people don't get these things off the bat. Still, they're fun to explain to people who don't get them.

Maybe we should have a thread! 

[bhodi]

[KallDrexx]

That is    on so many levels :D

[Lantyssa]

The sysadmin one is so true.

[Soln]

hahahah heheh meh uhhh   

[Cyrrex]

It still surprises me when people don't get these things off the bat. Still, they're fun to explain to people who don't get them.

Maybe we should have a thread! 

So I'm going to eat some crow.  Apparently, I never thought about it for any length of time because just now I gave it about ten seconds and had that "ding" moment somebody mentioned.  Seems rather obvious now...sorry for the derail.

[Engels]

F13's work is done 

[Margalis]

Many readers refused to believe that switching is beneficial. After the Monty Hall problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine claiming that vos Savant was wrong. (Tierney 1991) Even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy

Fun fact: I remember reading those Parade magazine things. It was fucking hilarious. You literally had PhD math guys writing in saying that vos Savant was completely wrong, it was so obvious, etc etc. It went on for quite a while.

Probability is often counter-intuitive. Really a lot of math is counter-intuitive, so it's often dangerous to rely on neat solutions or "common sense."

Here's a similar problem. Say you have three hotels in a row of different levels of quality. You stop by the first one and it's ok but you decide to move on and check out the second one. The second one is worse than the first. Should you continue on to the third hotel? (You can't go back - one way street!) It seems like the common sense answer is it doesn't matter, the third one could be worse or better, who the hell knows? But imagine the hotels are lettered A B C where A is the best and C is the worst. The possible situations you are in (second hotel worse than the first):

ABC
BCA
ACB

You could be in any of the three above scenarios. In 2 of them going to third hotel is a win. That really does not make intuitive sense but them's the breaks.

Also about Monty Hall - it doesn't matter if he knows which doors have goats behind them. If he opens the door with the prize behind it then you just lost and you can't switch so the whole question is moot. If he opens a door with a goat behind it you should switch.

The easy way to think about it is the chart of vos Savant's in that wikipedia page. There are only 3 possibilities! Just iterate through them. If you want you can add in cases where the host fucks up and reveals the prize, it doesn't change anything.

Also:

When I first heard this problem I thought as Cyrrex does now, then someone explained the million door version and I had that "aha" or lightbulb moment. It comes down to 1 of 3 ONLY if you dont change doors.

This is the single best strategy for wrapping your head around math and logic problems: consider degenerate cases. Change the numbers involved to 0 or 1 or a billion and what is happening is usually much more clear than using reasonable numbers. Also read this book:

http://www.amazon.com/Innumeracy-Mathematical-Illiteracy-Consequences-Vintage/dp/0679726012

Edit: Sorry for nerding it up with more off-topic talk, but as the child of a math professor and a match teacher I was biologically compelled to post.

[apocrypha]

as the child of a math professor and a match teacher I was biologically compelled to post.

Are you saying that mathematical ability is genetically inherited? Cos them thar's fightin words.