Monty's Dilemma

One of you I'm sure knows this classic puzzle:

I know the answer is you should always switch, but I can't remember the statistical reasoning behind it. Can someone remind me?

This depends on how the host decides what door to open.

The 'always switch' answer depends on the fact the host will ALWAYS open a door with nothing behind it. If only one door has nothing then he must choose that one. If both have nothing he can toss a coin to decide.

In this case the original probability is 1/3 for each door. After he shows an empty door the probability drops to 0 for that one, 1/3 for your door and 2/3 for the other one.

My favorite way to view this is to offer a different choice. The host does not open any door. Instead he gives you the option to keep the one you have or trade for both the others. This is essentially the same deal as before.

Sorry, I don't get that part. Why doesn't the probability drop to 1/2 for your door, 1/2 for the other door and 0 for the open door?

http://en.wikipedia.org/wiki/Monty_Hall_problem

Think about the 'trade for both doors' variant and how it compares to the actual puzzle.

Think of it this way, there are three possibilities:

1. You pick the winning box, and if you change you will lose.
2. You pick a losing box, and if you change you will win.
3. You pick a losing box, and if you change you will win.

2 and 3 are the same because there are 2 losing boxes and 1 winning box. You don't know if the box you have chosen is the winning box, but you are more likely to be in a situation where changing the box will result in a win, rather than preventing a win, simply because that possibility occurs more often.

Probability of first door being correct- 1/3. You picked 1 out of 3.
Probability of second door being correct- 1/2 as there are two options after the host eliminated one.

Thanks, I knew you'd all clear it up for me!

Is someone going for job interviews by any chance ?

Heh. No, a friend of mine was having a conversation at work about it, and he asked me...

this popped up in the 'curious incident about the dog in the night time' (or whatever the title was if it wasnt that)

you'd think that having 3 doors is 1/3 and then having one eliminated is going to be 50:50 but it just went on and on for pages upon pages of ok enough already who killed the f'ing dog still 1/3 probability

reminds me of the family guy blueharvest clip (probably on youtube like the rest of the show) "ok ive hidden the plans to the deathstar in one of these 20 boxes"

Like usual I have no idea what you are talking about. But my dog feels unsafe.

it was a book
and not a very good book
about someone with aspergers syndrome
(or was it asparagus i always get those two mixed up ;) )

boy finds dead dog
no one cares cept him and owner
boy writes it all up in the book you are reading along with aload of pointless why did i bother reading this book mathmatical problems

he even visits his estranged mum in london and instead of using the a-z and finding her street page and grid reference and working his route from there (or getting a taxi) his condition deemed he should start with the square hes on and work from there spiraling outwards

plot spoiler his dad killed the dog cos the dogs owner wouldnt get jiggy with him

it was a crap book that everyone seemed to be reading a few years back (before they found the davinchi code)

2 choices does not equal a 50:50 probability. You can either win or lose in the lottery, but the probability is not 0.5.

1 in a couple million.

1 in 14 million for the uk one iir

edit
must not type when using high res i cant see what im doing and misread the first quoite

afaik heads or tails is 50:50 thats 2 choices (a single coin toss that is not best of 3 or 3billion)

now IF you call heads or tails then its no longer 50:50 you have

call     side
tails    tails
tails    heads
heads    heads
heads    tails

In the lottery you do not have 2 choices, you have millions of number combinations to choose from, so your comment about the lottery not being 0.5 probability of winning is true, but irrelevant.

Ginger tea you still have 50:50 chance of winning after calling, as the other half of possibilities are excluded :S

play or dont play ;)

Curious Incident etc. was an excellent book.

Seconded.

My point was relevant if you did get it. The point I made was that if there are 2 different outcomes, outcome A="win the lottery" and outcome B="do not win the lottery", that doesn't mean they are equally probable. People make this assumption because when rolling a dice all outcomes are equally probable, and that's pretty much the only prob. math they've ever done. So in the Monty Hall's paradox there are 2 doors but the p to win the car is not 0.5 when you switch, even though you can either win the car or not.

I think that people tend to think it's 50/50 due to there being two doors not because of "win or not win". But the 50/50 is still flawed due to the previous condition of the game show host KNOWING what's behind the doors and then making a choice to expose a losing door, that's why it's 2/3 chance to win if you change.

The "there being two doors" is exactly the same point I was making.

In the Monty thing.

IF you choose the door Monty opened, then changing your choice to one of the remaining doors gives you a 50/50 chance of winning. If you had choosen one of the other doors to begin with, then changing your choice has no effect on the odds at all.

Your chances were 50/50 from the very start. There is no difference between a 1 in 3 chance where one is removed (and you are then free to choose between the remaining two) or one where you just choose from two to begin with.

If the original had been, you choose one of three, if your first guess is wrong then you can choose again, then your odds are really 2 in 3. Taken from the start. But calculated this way.

Original chance of getting it right roughly 33% (1 in 3)
second change 50% (1 in 2)
Your chance of winning is 88%


Wait a minute, got that wrong.

Your chance of losing the first round is roughly 66%
Your chance of losing the second round is then 50 percent. (or 33% of the original chance)

so your chance of winning at the outset is 66% 33% from the first and 33 more percent of the remaining original.

I explained it wrong but the 66% chance of winning is correct.

The probability of the two remaining doors is not equal, I now realise, because you have the extra piece of info that a losing door was removed after your first choice.

Chance first door wins=1 in 3
Chance second choice wins=chance first door loses=1- (1/3)=2/3

I've never seen the game show Monty so I'm not certain about what you are saying. But if anyone is still confused or does not believe that switching gives you the highest chance of winning, read the Wiki article posted earlier, it explains it well.

Pick a door, any door. What are your odds of having chosen the winning door? 1 in 3.

After you have chosen, with all doors remaining closed, if I offer you the contents of BOTH of the remaining doors, in exchange for your ONE door, does it increase your odds if you switch?

Sure. You've got a 2 in 3 chance of winning.

The game show host is effectively giving you both of the remaining doors, so take 'em.

The game show was called "Let's Make a Deal" and the hosts name was Monty Haul.

He offered 3 doors, the player picked one. Then he opened one of the two remaining showing that it had some junk in it. Then asked the player if they would like to switch to the other door (the one he did not open).

He is basically giving the player both of the other two doors. So, it makes sense to switch.

> He offered 3 doors, the player picked one. Then he opened one of the two remaining showing that it had some junk in it.

Did he always open a door that had junk? Or did that door sometimes have the prize and then the person did not win? What if the person originally choose the winning door? Different answers to these questions changes the possible results. (I only remember a little bit of the show, I was a "Price is Right" kind of guy)

By the way, the original TV game show was Let's Make a Deal.

Hmm, in that case I'll choose WIN. :D

Ah, but the question ignores the regret factor. If you chose the correct door the first time, and then gave it away by choosing to switch, you're going to feel worse than if you had the wrong answer all along but stuck with your gut.

It's like how if you go out fishing and don't catch anything you say, "Darn, I didn't catch anything, but hey, at least I got to go fishing." But if you go out fishing and have the biggest fish in the lake on your hook, only to let it slip out of your hands when you're pulling it up onto the boat, you're going to be mad that you let it get away.

Makes my head hurt thinking about it, but it makes sense... I like brain teasers like: What weighs the most? 50pounds of feathers or 50pounds of bricks?

although that one is easy.

i should have worded that one more clearly

if person a is making the call but person b is making the prediction of the call AND the outcome then its not 50:50
you could be right on the call and wrong on the outcome or right or wrong on both counts

if its just peron a and the coin then it is 50:50

same as if you predicted if a car (that isnt indicating) would turn left or right at a t junction (providing that the driver doesnt drive straight on that is ;) )

as for the curious incident book
some of it i enjoyed, the odd maths bit, but ive read better "i wish i had these books when i was at school" books. i may not understand alot of em (as my maths teacher felt it would be better to give her class calculators as early as possible so i cant do sums in my head or long division etc ... its calculator all the way) but i enjoyed the way the books were written as mini problem stories (pitty they are all in storage oop norf)

and other times i think the book is one of the worst uses of a rainforest going

Ah, but the question ignores the regret factor.

This is because the human brain is most certainly not designed for logical thinking :D

dynaman, he always opened a door with junk.


Ginger Tea, predicting whether another driver will turn left or right if they do not signal is still a 50/50 chance. So, that is a bad example.

it was an example of 50:50 chance ala coin toss so it was in that respect a good example of 50:50
(least he didnt open a door and show you a goat ;) )