Let's say i was on a gameshow, and there are 3 doors, 1 of them with a car, and 2 with nothing?

Question

My odds of getting the car right now are 33.3% right? Let's say i pick door number 1. The gameshow host opens, let's say, door number 3, he gives me a chance to change my choice. Let's say i do change to door 2, why do my odds change to 66.6% and not 50% when i change my door if there are only 2 doors left?

Answer 1

I did a big write-up of this a few days ago, but here's a simple way of looking at it :

If the door you initially picked had the car, and you switched to any other door, you'd lose.  That will happen 1/3 of the time, since your odds of picking the door with the car initially are 1 in 3.

If you picked any of the other 2 doors (with nothing), the host would open another door (with nothing, he's obviously not gonna open the door with the car behind it), and you'd switch to the winning door.  Your odds of picking a losing door initially are 2 in 3.

So, if you pick a random door and stick with it, your odds of winning are 1 in 3 (the odds that you picked the right door initially)...if you pick a random door and always switch after he opens one of the losing doors, your odds of winning are 2 in 3 (the odds of you picking a losing door initially).

Answer 2

The game show is "Lets Make a Deal" and it was huge in the '60s.  Your question was debated in papers like the NY Times, with different scholars taking different positions.  I'm not sure the questioned was ever fully answereed.

Answer 3

It is often called the Monty Hall Problem after the host of the game show Let's Make a Deal.

The correct answer is always to switch. It is counter intuitive but there is no debate on the answer. Always switch.  

It is because the the person opening the door knows which door has the good prize. Effectively combining the probability of the two unselected doors. If the host did not know where the good prize was and might actually open that door then the answer would be that it did not matter if you switched or not.

Answer 4

This does make for an interesting mental exercise, but the reality is your chance IS 50% when you decide to change, and not 66%.

The analysis which yields 66% depends on a calculation of past percentage.  But past percentage is always 0% or 100%.  In other words, it happened or it didn't, there are no 'chances' anymore.

To use the Monty Hall example, let's say I'm standing up there, I pick my nose, Monty thinks I picked a door, and so he eliminates one.

The odds of me picking my nose and him getting confused are astronomical, but the odds of me picking the right door are now 50-50.

Same scenario, but your Momma raised you better, so you pick door #2 and Monty eliminates one of the other 2 doors.

You have to ask yourself "do I stay or do I go now..."

The odds that door #2 is the right door are 50-50, regardless of whether you picked a door in advance or not.

It doesn't matter if you're standing on stage trying to wipe your finger on your pants, or standing in front of door #2, the odds of picking the right door is still 50%.

--------------------------------------...

I answered the question, I believed my answer, and I'm still not sure why .............. but I'm wrong.

There are only 2 'doors', the one you pick, and all the other doors combined.

Take an abstract example .... there are 100 doors, and after you pick, Monty eliminates ALL the other doors but one, and then you have to decide which door to take.

So there you are, knowing that you have a 1 in 100 shot at hitting the right door.  If you hit it, Monty's gonna randomly pick an empty door and keep it, then eliminate all the rest.  If you MISS, it's ok, because Monty's gonna tell you which door is the good one by getting rid of ALL the empty ones (except yours.)

So would you switch?

Of course you would ... with 100 doors, you assume you'll select an empty, and then just wait for Monty to tell you the right one.  On the 1 of 100 chance you actually select to good door, welllll, sucks to be you.

I hope this helps.

Answer 5

It depends on the situation.

After you pick a door, does the host PICK A DOOR THAT HE KNEW HAD NOTHING BEHIND IT?  If so, you should switch and increase your odds to 67%.

After you pick a door does the host PICK A DOOR AT RANDOM and he happened to get one with nothing behind it?  In that case it doesn't matter what you do, each door has 50%.

Try simulating the situation, that is the best way to see what does on.  Have someone be the host, put a prize in 1 of 3 bags, then choose a bag.  (You should both choose the bag with a die so you don't get influenced by anything that could throw it off. 1 or 4 means bag 1, 2 or 5 means bag 2, 3 or 6 means bag 3.)