Monty Hall Problem is a famous probability puzzle in statistics. It is named after Monty, the host of the television game show "Let's Makes a Deal"...
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Great explanation!
I think the
kdoors case is the best way to develop an intuition for the problem.Suppose there were 100 doors, you choose one. Then Monty goes ahead and opens 98 non-winning doors from the remaining 99, leaving one door unopened. I think the decision to switch makes more intuitive sense here, and then you can see that the intuition still applies when you scale back to 3 doors.
You are right! Since he never chooses that winning door so we increase our belief that is the one every time he opens the new door.
lol - I had a day-long argument with an ex-girlfriend about this. She insisted that it was a 1 in 2 chance of making the right choice after one door is opened and I was adamant it was 2 in 3. For once I was right :)
Anyway - it's a really good example of how people's intuition around statistics can be completely wrong. Same as the fact that one roll of a dice is never going to affect the outcome of the next...
Thank you. You can use my blog post as part of your explanation next time you see her. :D
I first learned of this problem about ten years ago and one of my hobbies for a while was "arguing" about it over a few drinks with my brother until he finally saw the light. I believe it was the hundred doors variant that finally got him to understand.
It's a strange coincidence that you posted this - I just wrote a Monty Hall Problem simulator on Thursday to get some Java practice in before my new job starts next week. It runs through a million iterations and gets a consistent 67% wins when switching doors.
I was really excited when my simulation result matches the exact result. This stuff is fun!
This is interesting: "This 'equal probability' assumption is a deeply rooted intuition (Falk 1992:202)"..."The problem continues to attract the attention of cognitive psychologists".
I actually had to wrap my mind around to find a good way to make sense of the "switching" strategy in the (n,k) version. Once my simulation worked, I gained some intuition and understood more about the original version.
It is weird how the brain works but it is good to know that I am not alone.
I'm sorry to point this out, but there is an error in your Bayesian formula.
(1/3)/(1/3+2/3) is 1/3 and not 2/3.
I'm not exactly sure where the error is.
I fixed it. Thank you!
written a python simulation when I was introduced to this and funny how true it was.