The Monty Hall Puzzle

[crokett]

You are a contestant on a game show. There are 3 doors. Behind 2 is a lump of coal. Behind the 3rd is a diamond ring. You are asked to pick a door. The host then opens one of the two you did not pick and reveals a lump of coal. The door he opens always has a lump of coal. Then he asks if you want to change to the remaining door or stay with the one you picked. Do you switch?

[rja]

I am a blacksmith. I could use that lump of coal. On the other hand, nothing trues up a grinding wheel better than a diamond. This would be a tough choice . . . .

[RayintheUK]

Switch doors - better odds of winning the diamond.

Ray.

[LarryG]

Doesn't matter whether you switch. There are now two doors left; your odds are 50-50 either way.

[LCHIEN]

I Agree with Larry.

[TheRic]

Doesn't matter whether you switch. There are now two doors left; your odds are 50-50 either way.

Yep! All you know is your odds went up from 30% (1 out of 3 doors) to 50% (1 out of 2 doors).

[DaveS]

This has been debated many, many times...

Here is the easiest explanation for why it improves your odds to switch.

The probability that you did not select the diamond is: 2/3 (67&#37. Once a coal door is revealed, the probability of winning the diamond is still 1/3 - however, if you switch doors, you win only if you did not initially select the diamond door (which is a 1/3 chance remember? 2/3 chances you did not select it). If you change doors, then you have a 2/3 chance of winning.

When I first thought about this, I assumed the 1/3 odds always prevailed. So, I wrote a program to iterate through thousands of trials, and I was intrigued to find that if you switch, you win 67% of the time, if you don't, you win 33% of the time.

This is an application of Bayesian probability.

[Russianwolf]

I concur.

If you lucked out and picked the diamond to begin with, the remaining door will have coal.

If you picked a door with coal, then the remaining door will have the diamond.


There is no way of knowing which it is until you open your door.

then again, there is so much pressure on the situation, it might be enough to convert the coal into diamonds and you'd win either way

[LinuxRandal]

I concur.

If you lucked out and picked the diamond to begin with, the remaining door will have coal.

If you picked a door with coal, then the remaining door will have the diamond.


There is no way of knowing which it is until you open your door.

then again, there is so much pressure on the situation, it might be enough to convert the coal into diamonds and you'd win either way

I was gonna ask what the time limit was, as time and pressure are neccassary for that conversion.

[cgallery]

Here is the easiest explanation for why it improves your odds to switch.

While your explanation sounds very statistician-like, including the reference to Bayesian probability, I just can't buy it. I'm afraid I'm going to have to call B.S. on that.

[DaveS]

While your explanation sounds very statistician-like, including the reference to Bayesian probability, I just can't buy it. I'm afraid I'm going to have to call B.S. on that.

I understand your doubt... been there...

Here are a few links to corroborate:

http://en.wikipedia.org/wiki/Monty_Hall_problem
http://www.uvm.edu/~dhowell/StatPage...ThreeDoor.html
http://www-cse.uta.edu/~cook/ai1/lec...threedoor.html

Here is a simulator so you can prove it to yourself...

http://nlvm.usu.edu/en/nav/frames_asid_117_g_4_t_5.html

[cgallery]

Here are a few links to corroborate:

"If you always switch from your initial choice, you will only lose when your first choice was a winner, so you should expect to lose 1/3 of the games. This means you will win 2/3 of the games."

Still trying to wrap my head around that.

[sacherjj]

Intuitively, you want to think that it won't make a difference. However, it makes sense that switching is the best option.

[crokett]

Dave has it correct. Let me 'splain. No there is too much. Let me sum up. The odds of picking the correct door initially is 1/3. This means the odds of not picking the correct door is 2/3. So after the host reveals the lump of coal and asks if you want to switch, you should switch since there chances are double (2/3 vs 1/3) that you did not pick the correct door to begin with.

[mater]

According to Dave's simulator I lost when I didn't switch and won when I did.