Probabilities from movie 21

Aikon

This is the problem that a teacher (Kevi Spacey actually) presented the class on the movie 21:
You are on a TV show and the announcer asks you to choose 1 from 3 doors.
Behind 2 doors there are a goat and behind 1 there is a new car.
The guy coose door 1.
The announcer go and opens door 3, in wich there is a goat, and asks the guy if he want to change.
He said yes and open door 2 and win the car.
Now the teacher asks why the guy changed the door? He answer, at the begining he had 33.3% of chance of being right, but when the announcer (that knew where was the car) asked if he would like to change, he thought that changing would increase his chance of winning to 66.6%.

My question is if this calculation is right? After 1 door was opened I think that probabilities would change to 50%.
What do you guys tell me?

Michael Redei

The calculation is correct; your chance remains 1/3 if you keep the door you selected first, and so the other door promises a 2/3 chance of winning the car. This problem (known as the "Monty Hall problem", which you can google) continues to cause discussions, since many people insist that two remaining doors should both have equal chances of hiding the car. The doors, however, are not equal, since you chose one (without knowing what was behind it), and the TV show host chose the other (knowing quite well what was behind).

Here's a similar scenario that might convince you that you should swap when the person running the game offers you the chance:

Imagine a lottery with 1 million tickets. One of these tickets is marked "$1 million", all others are worthless. First you choose one ticket at random. Your chances of having won the big sum are 1/(1 million), okay?

Now the person in charge looks at all remaining 999,999 tickets and throws 999,998 of them away. He's now left with one single ticket, which he offers you in exchange for the one you have. Will you swap?

If you now say that your chances are 50% of winning the money and you shouldn't swap, how do you explain the ticket that just had a chance of winning of 1/(1 million) having suddenly become more valuable, although you did absolutely nothing to increase its winning chances?

justsomeguy

Some implied rules about what doors the host can open before offering the choice are often left out of the example for this puzzle. They are:
1. The host cannot open your door.
2. The host cannot open the door with the prize.

The rules seem obvious in the context of a game show, but when mathematically figuring the probability people suddenly forget them.

If the host cannot open the door that reveals the prize, and 2/3 of the time you pick the wrong door initially, then 2/3 of the time the host has no choice about which door to open -- he must reveal a loser, because his other two choices are invalid -- your losing door, and the winning door.

If the host were allowed to open your door, then assuming he did so and revealed a goat, your chance of winning after switching would be 50%.

Michael Redei

There's nothing implied here. The host doesn't open my door, and he doesn't open the door with the prize either. What he could doesn't enter the matter.

If you don't bother about what has actually happened in this story, but you're going to study all possible scenarios, then you ought to consider the case in which the host opens a door and reveals the prize behind it. Now that's the kind of show I'd want to participate in.

HallsofIvy

It is a fairly easy computation to show that if the encee knows which door the prize is behind, and intentionally does NOT open that door, then, after a door is opened, the contestant can, by changing doors, make use of the emcee's knowledge to change the probability of winning from 1/3 to 1/2.

It is interesting that if we assume the emcee does NOT know which door the prize is behind but, by chance opens a door that does not have the prize, then switching doors does NOT change the probability.

justsomeguy

The change is from 1/3 to 2/3, not 1/3 to 1/2, as 2/3 of the time you picked the wrong door initially and the host had no choice about which losing door to open, as there was only one possible choice for him. This leaves the prize behind the unpicked and unopened door.

In this scenario can he pick the same door as you, or are his choices limited to picking one of the other two at random?

ImaLooser

I would say the odds change from 1/3 to 2/3.

? Are you certain? I don't see how it makes any difference what the host knows in this case. The number of choices is reduced, so the probability changes.

pwsnafu

Yes we are certain. See the Ignorant Monty section.

Edit: Oh wait. You mean it changed from 1/3 to 1/2. Then you're right.

scalpmaster

After the host reveal a door, a second contestant joins in to pick a door from remaining 2 doors shown.

(1) If he has seen the host revealing the door, is his probability higher than 1/2 if he always pick differently from the first contestant first choice?

(2) If he has not observed what happened earlier, does his probability change from 1/2 to 2/3 the moment he happens to pick differently from the first contestant first choice?

justsomeguy

If the 2nd contestant knows which door the 1st picked, and which one the host revealed the goat behind, then he has the same 2/3 chance of you if he picks the remaining door.

If the 2nd contestant knows which door the 1st picked, but not which door the host revealed a goat behind, his chance is 1/3. He only has two options:
1) Intentionally pick the same door as the 1st, and assume the same 1/3 chance.
2) Randomly pick one of the other two doors, a 1/2 shot at the remaining 2/3 -- i.e., a 1/3 chance.

Where does the 1/2 keep coming from? The scenarios that lead to a 1/2 chance are very convoluted.

scalpmaster

How can it be 1/3 if it is randomly picked? A new scenario with only 2 doors shown to a new player who has not observed anything should at least be 1/2 chance.

justsomeguy

Both of the doors he has to choose from could hold goats. 1/3 of the time, they both will. Therefore his chance of winning is 1/2 * 2/3, not 1/2 * 3/3.

scalpmaster

For this new player who has not observed anything, there is no longer a 3rd door in the choice equation (it can be hidden behind a curtain).
There are just 2 doors for him to pick from when he arrives.
He can randomly pick one of the two.i.e. Prob=1/2

justsomeguy

He can *pick* one with a probability 1/2, but the chance that the car is behind one of those two doors is only 2/3. The 1st door still exists, and still may contain the car. Even if you hide it behind a curtain. ;)

scalpmaster

Its already stated that the hidden door is no longer relevant since the host had shown it has no car. As far as the new player who has not observed anything is concerned, he is just presented with a NEW game where there are only 2 doors, one with the car. However, rather than guessing the 2 doors, the monty hall problem is saying the new player just need to bet that the first player is a loser.

justsomeguy

For some reason I thought it was the 1st contestants door you were 'hiding', not the one the host opened, sorry! It wasn't clear to me what you meant until now.

scalpmaster

So it is the info of which door first player picks rather than knowing which door host reveal that gives the edge from 1/2 to 2/3. The new player may not be aware but from the audience standpoint, is there some kind of a hidden transient change in the probability the moment he happens to pick the opposite as the first player by himself before the 2 doors are open?

Given a fixed time period,say 1 week, the stock market can go up,down or stay sideways.
A hedge fund can long, short or use rangebound(options strategy) to bet for that week closing level.

Hedge fund manager A opened long position the market on monday.
On tuesday, there is some major news/economic data that can be interpreted either way so the market will breakout of the trading range to move in one unknown direction(3rd door is out).

Hedge fund manager B comes along after the news. Should he
(1) randomly pick one direction?
(2) instead of analysing market direction, find out fund manager A's position and bet the opposite?