Monty Hall

[andrewducker]

[Poll #1770413]

Explanation

I have known what the answer was for ages, but for some reason it only "clicked" in my head today. You can blame sarahs_muse for triggering it.

Comments

[dpolicar]

Neither answer quite covers it.

I "believe" that switching is beneficial because of a combination of peer pressure, having found some argument that seems compelling that suggests it, and having sat down for an hour with a deck of cards and run Monty Hall simulations and counted the results.

But it has never "clicked" for me, which makes it a very unstable belief.

Then again, I have a lot of beliefs like that.

[dpolicar]

I should add that I also "believe" it on the philosophical grounds that probabilities (at least macroscopic linear ones) are quantifications of ignorance, not facts about the world, so it ought not surprise me that manipulating my ignorance will make things more or less probable.

But really internalizing that is hard.

[andrewducker]

I wrote some code this afternoon to prove it to myself.

What I discovered was that either I had chosen the prize with my initial guess, or I had not. Opening doors that I hadn't chosen did not change the correctness of my initial guess. And therefore, there was always a one in three chance that my original guess was correct.

Therefore, as there was only one other door to choose, that door must have a two in three chance of being correct.

[theweaselking.livejournal.com]

I don't like the answers because "I believe" is not correct. I *know* and can *trivially prove* that switching is beneficial. There is no belief required.

[andrewducker]

There is belief involved, but there is no faith. One can believe true things. In fact, on my philosophy courses the best definition that I encountered for knowledge was "justified, true, belief".

[theweaselking.livejournal.com]

I still don't like the term used here, as "believe" can also be in something untrue.

[andrewducker]

Yup. But as people on both sides would presumably think that their belief was true, using anything stronger would just have made people on the wrong side even more wrong :->

[theweaselking.livejournal.com]

I probably would have worded it as "switching is beneficial" versus "switching is not beneficial" versus "monty who?"

[andrewducker]

Yeah, that works better.

[spacelem.livejournal.com]

Bayesian statistics uses the phrase "prior belief" all the time.