Explanation of the Monty Hall problem paradox?

Question

You have 3 doors (door A, door B, door C). Behind one of the doors there is a car. There is a goat behind the other two.

First trial:
You pick door A. The gamemaster opens door C disclosing a goat. According to Monty Hall’s theorem, you choose to open door B to improve your probabilities from 1/3 to 2/3 to get the car.

So, here we have door A with a probability of 1/3 and door B with probability of 2/3.

Second trial:
You pick door B. The gamemaster opens door C disclosing a goat. According to Monty Hall you now choose to open door A to improve your probability from 1/3 to 2/3 to get the car.

So, here we have door A with probability of 2/3 and door B with probability of 1/3.

So, the Player has influenced the probability of getting the car behind a given door simply by the sequence of his first picking a door.

Can you explain this paradox? Best answer will be awarded to the clearest and most convincing explanation.

Answer 1

Normally all three doors would have an equal chance of having the car behind it.

When the game master opens a random door (different from what you selected), this does NOT change the odds AT ALL.

1: the game master shows a car. You know you lost, whether you switch or not
2: the game master shows a goat. You have a 1:2 chance of getting the car; either the car is behind the door you chose, or it is behind the other door.

But if the game master shows you a goat, you should change your choice. Why? Generally speaking the game master wants you to get the prize, as this is good for business. The game master knows which door contains the car. If you have selected the correct door, he would have opened the door you selected. If you have selected the wrong door, he will open: not your chosen door, not the door with the car, thus the third option.

Answer 2

The Monty Hall paradox boils down to simply this idea; the door that is revealed are not necessarily randomly selected. For the first pick, one of two things could happen:

1. Suppose you picked a goat on the first try (probability is 2/3). Then, because one goat is left, that door would be revealed next. If you switch, you get the car; if you don't, you get a goat.

2. Suppose you picked the car on the first try (probability is 1/3). Then, one of the two goats are revealed. If you switch, you get a goat; if you don't, you get the car.

Now look at this in terms of switching or not switching instead of getting a car or a goat the first time. This means switching gives you a 2/3 chance of getting the car (from 1.) and a 1/3 chance of getting a goat (from 2.). If you don't switch, you have a 1/3 chance of getting the car (from 2.) and a 2/3 chance of getting a goat (from 1.).

Answer 3

The explanations you have seen so far are pretty good. It can be difficult to figure out, due to the 1/3 odds. With such high probabilities, the Monty Hall question may not fully make sense.

When someone reads the explanation and still cannot see it, I propose a new Monty Hall question.

Lay out a deck of cards face down. I ask you to pick the Ace of Hearts. You choose a card at random. I then show you 50 cards where the Aces of Hearts is NOT. You now have a new choice: You can stick with the card you originally chose, or you can choose the other unrevealed card.

When you expand the question to a much higher number, it becomes more obvious why you always have a 1/3 chance of being originally right instead of 1/2. With the deck of cards, you only have a 1/52 chance of being right. When I expose 50 cards and give you that choice, your odds do not mysteriously change to 1/2. The probability of you being right is still 1/52.

It took a while for me to sort this one out. The deck of cards example really helped me.

Answer 4

You have 3 doors, one is a gag.

`You FIRST choose one of the doors. Doesn't matter if you choose right or not, your odds of getting the prize is 1/3.

Monty will ALWAYS offer you another choice, and it will NEVER be the prize.

Unless you choose the right door right away, you start out with a lesser choice. If you keep that choice you will have 1/3 chance of winning the car on your first pick.

If you switch you are taking another of the 3 chances giving yourself another chance at the prize for a total ot 2 of the 3 chances....2/3.

Quick and dirty explanation I know, but if you want a more full and detailed explanation....just send me a message with the link to this question in it.

Hope this helps.

Answer 5

You are provided with information when one wrong answer is revealed. At first, you have a 1 in 3 chance of picking the right door and a 2 in 3 chance of picking a wrong door. When one goat is revealed, you are left with a goat and a car, and if you change doors, you invert your outcome. The odds from your intial guess are unchanged. You had a 2 in 3 chance of a goat and if you change you get the car. You had a 1 in 3 chance of picking the car and if you change you get the goat.

Answer 6

I may be missing something, but this doesn't make sense to me - the odds are 1 in three when you first pick. The disclosure of a goat behind one of the doors, makes the odds now 1 in 2 for each of the other doors. If you open one of the other doors, the odds are now 1 in 1, or certainty. No other odds influencing occurs, that I can see. Am I missing something?

Answer 7

I saw this in "ask Marilyn" Here is the answer:
http://www.physics.uoguelph.ca/tutorials/sig_fig/SIG_dig.htm

Basically, if you choose one door, you only have a 1/3 chance of winning. There is a 2/3 chance that one of the other two doors will be the winner, and the game master just helped you out in telling you which it was NOT!

Answer 8

the idea that the door you picked has less a chance of containing the winning prize then the other door not opened in ridiculous..
the chance is equal for both... first it was 1 in 3 ... when one door is opened, it still is 1 in 2 for both doors....
just like deal or no deal, the chance you have ANY certain case is 1 in 26 ....
regardless of how many cases are eliminated, your initial chance of picking the winner never changed... just the CURRENT odds change as cases (doors) are exposed.

Answer 9

Go to www.wikipedia.org for the answer. That's how I figured it out.