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The Time Everyone “Corrected” the World’s Smartest Woman (2015)

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Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#111

Post by spenczar5 »

TchoBeer wrote: post #64
dllthomas wrote: post #34

Earlier quoted context omitted.

Interestingly, if the host picks randomly (and if he reveals the car, you... start over, or you get the car, or you get nothing, or ... it doesn't matter because it happens not to have happened in the time we're considering) then you are faced with a 50/50 chance.

I don't understand how this could be. If you start over when the host picks the car, then isn't that the same as the host picking the goat every time, i.e. the same as the host knowing.

This is a small enough situation that we can do complete enumeration of the states.

Let's say you pick door 1. Let's go through all the possibilities: the states of the 3 doors, which one monty reveals, and what do we do, and what is the result?

  1 | 2 | 3 || Monty Reveals | You switch | Result
  ---------------------------------------------------
  G | G | C ||      2        | Yes        |  Win
  G | G | C ||      2        | No         |  Lose
  G | G | C ||      3        | N/A        |  Start Over
  
  G | C | G ||      2        | N/A        |  Start Over
  G | C | G ||      3        | Yes        |  Win
  G | C | G ||      3        | No         |  Lose
  
  C | G | G ||      2        | Yes        |  Lose
  C | G | G ||      2        | No         |  Win
  C | G | G ||      3        | Yes        |  Lose
  C | G | G ||      3        | No         |  Win
Count 'em up: When you switch, 2 wins and 2 losses. When you don't, 2 wins and 2 losses. Of course, the situation is symmetrical for any starting guess you make.

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#113

Post by mathrando »

6gvONxR4sf7o wrote: post #13

There are a number of things I like about the monty hall problem. There's the history, the unintuitiveness, the subtle easy-to-screw-up nature of probability problems, the calculation, the sociology, and the overconfidence of wrong experts. Most of all, it's the calculation vs intuition that I like. You can do the calculation, or run simulations and prove correctness. In fact, it would be much harder to be so widely…

Whether Monte Hall is counter intuitive is a function of how the question is phrased.

When you phrase it in a way that underlines the mechanical nature of the host's decision, people get it right. When you phrase it in a way that suggests the host's choice is itself random, people get it wrong.

I think the first formulation primes people to think of it from the perspective of the host, which is the right perspective for this problem.

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#114

Post by FabHK »

Ok, so this is about the Monty Hall problem. A few remarks:

1. It is crucial exactly how the problem is framed. The "correct" framing is:

a) Contestant choses a door, b) Monty Hall will pick a door (different from the one the contestant chose) SUCH THAT a goat is behind it (if there are two, pick one randomly), c) contestant is offered a choice to switch. In that case, the correct answer is to switch, increasing the chance for a car from 1/3 to 2/3.

(The alternative framing is this: a) Contestant choses a door, b) Monty Hall picks a door randomly (different from the one the contestant chose), and if a car is behind that door, then the game ends, otherwise, c) contestant is offered a choice to switch. In this case, switching doesn't matter, the chance of getting the car is indeed 50/50, switching or not.)

2. Not everyone "corrected" her, some people did, and some people supported her. The confusion arose, arguably, because of the unclear framing. (One could argue that the phrase "the host, who knows what’s behind the doors" hints that the first framing is intended, but why not make it explicit? The host could know what is where and still choose randomly, in which case we're at the alternative framing.)

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#115

Post by ronyeh »

lachstar-x wrote: post #102
6gvONxR4sf7o wrote: post #13

There are a number of things I like about the monty hall problem. There's the history, the unintuitiveness, the subtle easy-to-screw-up nature of probability problems, the calculation, the sociology, and the overconfidence of wrong experts. Most of all, it's the calculation vs intuition that I like. You can do the calculation, or run simulations and prove correctness. In fact, it would be much harder to be so widely…

Can you help me understand - in my view the choice to switch doors or keep the same door is irrelevant, because even if you keep the same door you're making a choice that is now 2/3 of the right answer. If you switch or keep, you're still choosing from two doors that contain a car and a goat. The other door is no longer relevant and doesn't affect the new state at all. It's your perspective (narrowing the choice down…

Think from the perspective of your first choice. You had a 66.66% chance of choosing the wrong door.

So now I just eliminate one of the doors you didn't choose. There are two left, including the one you chose first, which only had a 33.33% chance of being correct. Nothing else has changed about the problem.

What if we started with a thousand doors? You chose 1, with a 999/1000 chance of being wrong.

Of the remaining doors, I open 499 doors. The prize is behind one of the remaining 500 that you did not choose the first time. Do you want to have a chance to pick from one of the 500?

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#116

Post by hn_throwaway_99 »

6gvONxR4sf7o wrote: post #13

There are a number of things I like about the monty hall problem. There's the history, the unintuitiveness, the subtle easy-to-screw-up nature of probability problems, the calculation, the sociology, and the overconfidence of wrong experts. Most of all, it's the calculation vs intuition that I like. You can do the calculation, or run simulations and prove correctness. In fact, it would be much harder to be so widely…

In this case, though, I don't believe it's just calculation vs. intuition. The reason the 100 or million door problem helped me understand was because it clarified an implicit rule of the game: that the door the host chooses to open is not random. If it were random, it would make for a very boring game ("Uhh, there's the prize, you win, I guess that's game over.") Once I understood that the host would only ever open "goat" doors, that's when I better understood the argument.

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#117

Post by TacticalCoder »

lachstar-x wrote: post #102
6gvONxR4sf7o wrote: post #13

There are a number of things I like about the monty hall problem. There's the history, the unintuitiveness, the subtle easy-to-screw-up nature of probability problems, the calculation, the sociology, and the overconfidence of wrong experts. Most of all, it's the calculation vs intuition that I like. You can do the calculation, or run simulations and prove correctness. In fact, it would be much harder to be so widely…

Can you help me understand - in my view the choice to switch doors or keep the same door is irrelevant, because even if you keep the same door you're making a choice that is now 2/3 of the right answer. If you switch or keep, you're still choosing from two doors that contain a car and a goat. The other door is no longer relevant and doesn't affect the new state at all. It's your perspective (narrowing the choice down…

> The other door is no longer relevant and doesn't affect the new state at all.

There is no new state, that's the thing. By picking one door you split these in two groups: one group with 1/3 to win, the other with 2/3 to win. When the host reveals the door with a goat behind it from the group that is 2/3 to win, it doesn't change the fact that that group had 2/3 to win to begin with.

I mean: that's how I see it.

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#118

Post by phamilton »

The simplest explanation I've heard that has helped it click for people is "You can choose Door A, or both Doors B and C". Clearly choosing both B and C is a 2/3 probability. Then you guide them to understand how the host revealing one of B or C doesn't detract from the fact that you are still choosing both B and C to begin with. And finally establishing that switching is equivalent to choosing B and C instead of A.

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#119

Post by hervature »

dllthomas wrote: post #34
javajosh wrote: post #23

So this is at least in part about the "Monty Hall Problem" and why it's solution not intuitive. The article missed an important angle: when the host opens a door, he's giving you more information , which explains why it's better to switch. If you're the host, you need to know which door the car is behind to do your job 2/3 of the time, to avoid revealing it. It's this quality of unexpected information exchange that I…

Interestingly, if the host picks randomly (and if he reveals the car, you... start over, or you get the car, or you get nothing, or ... it doesn't matter because it happens not to have happened in the time we're considering) then you are faced with a 50/50 chance.

This is exactly why the Monty Hall problem doesn’t apply to Deal or No Deal. Intuitively, it disappears when doors are opened randomly because, by Bayes, the likelihood you picked the good prize goes up every time you reveal a bad prize.

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#120

Post by mjmahone17 »

saeranv wrote: post #79
GlenTheMachine wrote: post #38

Earlier quoted context omitted.

This particular one isn’t that hard to just do by hand. Which I did at the time, was surprised at the result, and stared at the column long enough to convince myself why she was right.

Using Bayes theorem? But if you know to use Bayes theorem, then you're already formulating the problem correctly, which is the problem all the critics are running into. Edit: Someone mentioned you could manually "simulate" it with a pen and paper, which I think is what you meant. If so, very good point.

You can brute force this without knowing Bayes’ theorem (see the grids in the article as an example of iterating through every possible solution then counting to arrive at the right probability).
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