[fool]

The chances were always 50/50 because by design the host will eliminate an empty curtain, so by design there's only two curtains, one of which has the car.
The stuff about the host and the curtain has no more impact on the odds than the host putting on his hairpiece or not, it's just part of the show leading up to a choice between two curtains.

[zippy2006]

Originally Posted by fool

The chances were always 50/50 because by design the host will eliminate an empty curtain, so by design there's only two curtains, one of which has the car.
The stuff about the host and the curtain has no more impact on the odds than the host putting on his hairpiece or not, it's just part of the show leading up to a choice between two curtains.

Get out your peanuts
and
you will find you are wrong

[zoo22]

Originally Posted by fool

The chances were always 50/50 because by design the host will eliminate an empty curtain, so by design there's only two curtains, one of which has the car.
The stuff about the host and the curtain has no more impact on the odds than the host putting on his hairpiece or not, it's just part of the show leading up to a choice between two curtains.

Seems like it, but no.

Here's a very simple way to test it for yourself with a deck of cards (you can do it with three cards, more closely simulating the game, but it takes a lot longer to see the pattern, while this ought to take about 20 seconds):

Curtains = Cards
Winning curtain (car) = Ace of Spades
Losing curtains (not car) = Non-Ace of Spades cards

- Pick a card from a full, shuffled deck. You want the Ace of Spades.
- Look at the card. If you didn't choose the Ace of Spades, you should have switched.

Here's what's going on:

- You choose a random curtain (you choose a random card)
- The host knowingly shows you an empty curtain, leaving one (someone shows you 50 non-AoS cards, leaving one card)
- You're offered a chance to switch curtains (you're offered to switch your card for the one remaining card)

You can obviously do it with someone else, and actually have them look through the deck and show you/discard 50 non-AoS cards, but all you really need to do is pick a card and see if it's the AoS. If it's not, then you should have switched.

Originally Posted by fool

by design the host will eliminate an empty curtain

I'll point out that it's precisely because of this design that it helps you. The host has to knowingly be opening an empty curtain for you. Otherwise, is it's 50/50 and it doesn't matter if you switch.

The host must know what's behind the curtain for the switch to matter.

[zoo22]

Originally Posted by rstrats

zoo22,

re: "Yes, it does matter if that the host knows what's behind the curtains."

It makes absolutely no difference if the host does or doesn’t know where the car is other than the game being prematurely over if he picks the curtain with the car. As long as the game continues, though, you should always switch. You can prove this by doing the experiment suggested by The Graphite, .

This is why I've grown to love the "does it matter if he knows" question more than the "should you switch" question.

Yes, it does matter if that the host knows what's behind the curtains.

It's not really the math that interests me most; it's our behavior. Once you know the answer to the difficult and counter-intuitive "should you switch" question, it becomes just as (if not more) counter-intuitive to accept that it matters whether or not he knows. People's minds just don't like it (either of the questions).

It's also interesting to me because I think most of us don't really "understand" our problems or even the answers very much at all. Often, we just think we do. Note I say "we."

Pigeons, by the way, have a better record with the "should you switch" problem than humans do:

To shed light on why humans often fall short of the best strategy with this kind of problem, scientists investigated pigeons, which often perform quite impressively on tasks requiring them to estimate relative probabilities, in some cases eclipsing human performance. Other animals do not always share the same biases as people, and therefore might help provide explanations for our behavior.

Scientists tested six pigeons with an apparatus with three keys. The keys lit up white to show a prize was available. After the birds pecked a key, one of the keys the bird did not choose deactivated, showing it was a wrong choice, and the other two lit up green. The pigeons were rewarded with bird feed if they made the right choice.

In the experiments, the birds quickly reached the best strategy for the Monty Hall problem — going from switching roughly 36 percent of the time on day one to some 96 percent of the time on day 30.

On the other hand, 12 undergraduate student volunteers failed to adopt the best strategy with a similar apparatus, even after 200 trials of practice each. Full article

LiveScience: Pigeons Beat Humans at Solving 'Monty Hall' Problem
Discover: Pigeons outperform humans at the Monty Hall Dilemma

Both interesting articles (I think).

[Spitfire]

Yes, it matters that the host knows. Because, as I've tried to point out, the host will be forced to reveal that one of the options is not what you want.

Now, if you play this game yourself, not knowing, and if your first guess is wrong, then the odds become 1 in 2 that your next guess is correct.