monty hall – how it works

I was lying in bed last night, wondering how the Monty Hall trick works, and finally figured it out by twisting around my perception of the problem.

What is the Monty Hall problem? Consider three doors. Behind one is a car, and behind two others are goats. You are asked to choose one door – if it’s the one with the car, you win. otherwise, you lose.

you choose one car. Then Monty (the gameshow host) opens one of the other doors to show a goat behind it. You are given the chance to change your choice of door to the other closed one – should you?

The answer is Yes. the /intuitive/ answer is that there is a 50% chance either way, but mathematically, there is actually a 66% chance.

Took me a good few minutes to figure out how to verify it. I kept thinking of it as “there is a 33% chance of me picking the right one. door opens. I now have… 50%?”. Couldn’t seem to make the leap for some reason.

That was a result of wrong perception – you need to think of it from the point of view of what is /not/ the right door.

  1. choose one door. the chance of the car being behind one of the other doors is 66%.
  2. one of the other doors opens. the chance is still 66%!.
  3. You now have two closed doors. the door you /did not originally choose/ has a higher chance than the one you did choose, so you should switch.

7 Comments.

  1. Is this a joke?

    You have two closed doors. One has a car, the other doesn’t.
    That’s a 50/50 chance.

    I understand the way your trying to view it, but it doesn’t hold water because your not dealing with the fact that you are no longer dealing with three doors, your dealing with two.

    At the beginning each door had an equal chance of having the prize. Now that only two remain, they still both have an equal chance.

  2. @Robert, nope – no joke. wikipedia describes it better than I do, including diagrams. It’s a head-twister, but totally true. Still hurts my head to think about it.

    Possibly the best way to verify this for yourself without doing any maths is this: Imagine there are 1000 doors. You choose 1. The host opens 998 of the others to show that they are goats. Now, what is the chance that you chose the correct door in the first place…

  3. I’m still grappling with this too. I think there is a missing piece here or something.

    Chance car is behind any door, is 33.3333%
    (or) the chance of picking a goat is 66.6666%

    If the host opens one of the other doors to reveal a goat, the new odds are:

    50% chance door reveals a car, or 50% chance door reveals a goat.

    Now, *if* the rules were: Host will open up 1 door out of the 2 doors that contain goats (at random) e.g. it could actually be the door you picked!… then, yes, the odds are a bit different.

    If I chose a goat door, there is a 50% chance that my door would have opened… the fact that it didn’t… *might* indicate to me that I don’t have a goat (and that it wasn’t just luck)… thus I’d actually be inclined to keep my door selection.

    Then again, $10 says I wouldn’t gamble on it.

  4. @Steve – your explanation has hurt my head. I think I may need to purge it with a nice pint and maybe a bout of “lalala” with my fingers in my ears.

    how’s this then:
    In the beginning, you choose a door. The chance of the car being behind one of the others is 66%. The host knows that one of the other doors is a goat. You know that as well. Knowing that does not alter the odds. The host proves that one of the doors is a goat by opening it. The odds still /have/ changed. Instead of it being a 66% chance that the car is behind one of two doors, it is now 66% chance of being behind one door (however, it is still 66% chance that it is /not the one you chose/). You choose to swap because it would improve your chance.

  5. Pictures are worth 1,000 words! I looked at the pictures on wikipedia, and now it makes sense.

    There are 3 possible arrangements of car vs. goats, after the host reveals the goat door, for switching, 2 scenarios turn up a car, 1 turns up a goat (66% chance of car if you switch)

    now my head really does hurt. thatnks for the brain teaser… sorry to doubt the answer too!

  6. Consider this. You pick a door. At this point it’s 2/3 that you’ve chosen a goat.

    The host opens one of the remaining doors, revealing a goat.

    Are the odds still 2/3 that your door has a goat while the remaining door has the car?

    Answer: it depends on the host’s state of knowledge.

    Ask him: “When you opened that door and revealed a goat, did you know it was a goat?”

    If his answer is yes, the odds are 66% the remaining door has the car.

    If his answer is no, the odds are 50/50.

    Hope that makes sense.

  7. the easiest way to see it intuitively is this:
    imagine that you initially choose one door from 10,000 doors. after you choose your door the host opens 9,998 “goats” doors and leave you with your chosen door and one more closed door. do you still think it is a 50-50 chance that the other closed door is the car door?

    hope my comment made sense…

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