The Monty Hall Problem

Inspired by Ramesh' puzzler

You have won a quiz show, and it's time to collect your prize. There are 3 boxes. One contains £1,000,000. The other two are empty. You choose a box (any box). The host of the show then opens another box to reveal that it's empty. You then have the choice between sticking with your first option, or switching to the other closed box.

Do you
a) Stay where you are
b) Switch to the other one

and why?

 

I'd stay where I was, because if I'm in a store, and change lanes, it always takes longer than if I'd stayed in my lane. So I'd stick with the box I was at.

 

Get a gun, and make SURE you get the right box.

 

I'd stick to the same box cos it's like that Schrodingers cat theory isn't it... it's actually in a state of flux between the two boxes until you open one

the explanation... kindof

Schrödinger's cat and Superposition

When studying quantum mechanics, there are two possible descriptions of a particles existence, the classical level and the quantum level. The classical level is that which tells us about the macroscopic things within the classical world, events and objects that we are more immediately aware of. This is the level on which we can witness events using ordinary ideas of probability. The quantum level explains those objects in the quantum world - molecules, atoms, subatomic particles and so on. Due to their small size, this world is often believed to be relevant only on a small scale.

A theory which discusses these objects is Schrödinger?s equation, which shows that any particle in the quantum world can be in two different states at the one time. An example of this is a radioactive atom which can be in both a state of decay and a state of non-decay. When we try to magnify information from the quantum world to the classical world, the particle?s probability amplitude is converted to a classical world probability. This collapses the wave function, causing only one of the states to exist.

Erwin Schrödinger proposed the "Cat in a box" experiment in 1926 as a way of explaining just one of quantum mechanics? many peculiar notions - that matter is able to exist in more than one state or position at one time, and it is only measurement, i.e. observation, that collapses this superposition.

Imagine a cat in a box. Nice. Now seal the box and imagine the cat comfortably sitting next to an unopened flask of poison gas. Attach to this scene a radioactive which will decay after a certain amount of time (measured by the half-life of the atom). This will cause the flask to open and set the gas free. Assume that the decay of the atom is completely random - that is, there is a fifty-fifty chance of the atom decaying within the time limit of the experiment (one hour) and causing the cat to be poisoned and killed. This would mean that there is a fifty-fifty chance that the cat is either dead or alive.

You can try to guess whether the cat had died yet, but you really have nothing to base it on (other than that strange smell?). The life of the cat depends on the decay of the atom, if the atom has decayed, the cat will be dead, if not, the cat will live. Schrödinger?s equation shows that an atom can be in both a quantum state of non-decay and a quantum state of decay at the same time, and thus the cat is in a quantum state of being alive and a quantum state of being dead at the same time. This is how we justify saying that the cat is both alive and dead. Until the box is opened to see what has happened, a wave function for both possibilities of the atoms state exists. Once the box is opened the probability amplitude of the particle converts to a classical world probability, collapsing the wave function, and causing the atom (and thus the cat) to be in either one state or the other. That is alive or dead in the case of the cat, and decayed or not, in the case of the atom.

In summary, one of the features of quantum mechanics is that objects (e.g. subatomic particles) are able to be in different states of being at the same time. The cat is the observable property of this experiment, and until observed it is in a state of being both dead and alive. According to quantum theory, until the system is measured, the radioactive atom is also in a superposition of being both decayed and not decayed at the same time. Erwin Schrödinger used this "cat in a box" situation, along with a whole load of quantum theory as a means to explain the notion that, under certain circumstances, matter can exist in more than one position or state at a time.

 

DQ

I do appreciate the reply you gave. (Quantum physics, and Schrödinger's cat, and Heisenberg theories about stuff at the molecular level fascinate me), but this problem doesn't concern that. This can be solved purely by thinkly logically on the "real" level.

Here's a clue. Think about what would happen if you originally chose the correct box. This should help.

 

It's still 50-50. Whether you had orginally chosen the right box or one of the wrong boxes, the host could still have opened an empty box, as he presumably knows where the money is.

All that has happened is one box has been removed from the game. You have not been supplied with any new information. So it doesn't matter whether you stick to your original box or not.

50-50.

 

Thanks for the reply. Yes, the host does know where the money is, but unfortunately, the answer is not 50-50. You need to think a bit harder

 

Thanks for the reply to my reply (why are you talking to me like you're my maths teacher??).

Anyway, I still reckon you're wrong. I'd certainly be interested to see any answer that suggests its anything other than 50-50.

 

// points and laughs @ OJK

 

I'd guess it's something to do with the fact you've already picked one box....

 

OJK's report card.

B+ for effort
C- for brains
E for attitude.

Summary: John is a good student who has the potential to do well. However, he does not apply himself as best as he could and always thinks he knows best. Even when informed that he is wrong.

 

You should stay...the chances of you picking correctly in the first place are probably higher than the chances of you picking incorrectly but then being given the chance to change and then picking the correct box.

Or something.

Possibly, something else altogether.

 

It's definitely not 50-50. I used to think it was and spent days arguing with my maths teacher over it, until he convinced me of the right answer.

Here's a clue. What happens if the first box you choose is empty?

 

Well, if the host then opens the box that you picked and it's got the money then you can say that was the one you picked for him to open.

but if it's empty you can say it was the one you picked for him not to open.

it's all about forcing choice.


heh, well, it kinda made sense to me.

So it's kinda 100% really. They don't tell you to pick a box you think the money is in or one you think it isn't in.

 

It's in the other box?

 

You walk away disgruntled?!?

 

Alpha.

What?

 

What happens if the first box you choose is empty?

According to the original question, if the first box you choose is empty, the host opens the other empty one.

Result: one open empty box, and you do not know if your original choice was right or not.

If you choose the right box, the host opens one of the empty boxes.

Result: one open empty box, and you do not know if your original choice was right or not.

 

heh, well, it made sense to me.

It doesn't tell you to try and pick the box with the money.

 

According to the original question, if the first box you choose is empty, the host opens the other empty one.

Exactly, so you agree that if the box you choose first is empty (whether you know it or not), then you should switch to win the prize

 

LOL @ "Alpha - what?"

 

Alpha's post made sense to me. Halibut's never.

 

Exactly, so you agree that if the box you choose first is empty (whether you know it or not), then you should switch to win the prize

Certainly. Unfortunately, I do not know if the box I chose first was empty.

 

ok


3 boxes


1 opened and empty.


therefore 2 boxes left.

1 box cash
1 box empty

Probability is 50:50 blind choice.

I neither stick nor choose the other box because I havn't won a quiz show.

 

(whether you know it or not)

Not that that's a crucial piece of info...