why does probability change in the monty hall paradox?

Question

first, do not attempt to re-explain the question; i already read it more than once from different books and sources.
second, do not copy and paste what you see off the internet but give an answer of your own understanding, or else leave it.

here goes:
suppose there are 3 doors, behind 1 of which is a gift you want, the other 2 are goats. you choose one, and the host reveals to you a goat from one of the 2 remaining doors.

originally, your chances of choosing a goat was 2/3, and chance of the gift, 1/3.
now, your chances of a goat is 1/2, and chance of the gift, 1/2.

why does probability change when the settings are all the same?

Answer 1

The opening of one door improves your chances by removing one of the wrong choices. Think about it this way:

Instead of 3 doors, have 100. If you pick from this set, there is a one percent (1/100) chance of selecting the right door. But opening one door removes one of the wrong choices. If you then choose from the new set, your chances have improved, and are now 1/99.

Answer 2

No the probability doesn't change. As what you had said, your chance of choosing a goat is 2/3. Which means its more likely than not that you had choosen a goat, and also means that the chances of some other door having the gift is 2/3. When the host reveal the goat, the chances do not change. It still means that 2/3 of the time you had chosen a goat and 2/3 of the time the other door has the gift.

The chances of staying with the door now only has a probablity of 1/3 bearing the gift while the chances of switching to the other will give you a probablity of 2/3, and not 50-50 as you had said.

This can be easier explained on a variation of a game. Let's say they give you 1000 doors and will still expose the goat and then allow you to make another choice, which they then repeat untill there's 2 doors left. The probablity of any door holding the gift is 1/1000, being almost impossible to have the gift. To maximize you winning, choose any door, then stick to that door until there is only 2 doors and then switch. This way, the host will foolishly open 998 doors with goats. Therefore, the last door remaining would have 999/1000 chance of bearing the gift, while your own door still only bear 1/1000 chance, and I think you would agree that the door you first choice is almost certain not to have the gift.

What really happens is that when the host reveals a door, the chances of that door is now spread out equally on all the doors that you had not pick. The probablity of the door you picked will not change because it is not possible for the host to open it, and hence it cannot get its chances raised.

Answer 3

The probability does NOT change.

You pick a door and you have a 1:3 chance of a prize. There is in fact a 1:3 chance of a prize behind EACH door .

So there is a 1:3 chance you chose the prize and a 2:3 chance that you did not.

If one of the doors you did not choose is opened and reveals a goat then necessarily there is a 2:3 chance that the prize is behind the remaining closed door (the chance it is behind yours is still 1:3 remember, and the total must add to 1:1).

So you double your chances by changing your mind given the additional information of the open door.

The chance is NEVER 1:2 behind any of the doors.

Answer 4

There is no paradox, there is a paragoats, haha a little humour
when you first chose odds were 2/3 that you would pick a goat
IF you CHOOSE AGAIN then you are working with new data
one door is eliminated and choice of two doors makes odds
1/2 if doors were closed and prize and goats shuffled
then odds of picking goat return to 2/3

Answer 5

No . . . the finished using mechanism in the back of the Monty hall paradox is that the alternative made with the help of removing between the non-triumphing possibilities is NON-RANDOM, e.g. the agent offering the alternative is particularly-known with it particularly is the perfect door. If the guy with the envelope does no longer understand this suggestions, then her strikes can't strengthen the prospect of triumphing.

Answer 6

you can think of the problem this way:
when the host opened a door for you you can consider a change in the settings (theoretically)
and this lead you to another probability space
where we only have tow doors and actually the chance in this space to open the right door is 1/2
but practically we still have three doors and the chance to open the right one is 1/3 so it depends on your way of dealing with the problem to say the truth
and 1/2 is true and false in this case
and we can't tell what is it really ..