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The Time Everyone “Corrected” the World’s Smartest Woman (2015)

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Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#101

Post by alew1 »

HALtheWise wrote: post #85
TchoBeer wrote: post #64

Earlier quoted context omitted.

I don't understand how this could be. If you start over when the host picks the car, then isn't that the same as the host picking the goat every time, i.e. the same as the host knowing.

Fair warning: I haven't rigorously verified the math on this, and tfa is all about the importance of vigorously verifying the math. In the case where the host revealing the car means you restart, there are three equally likely situations after the host opens a door: - You picked the car, and the host revealed a goat - You picked a goat, and the host revealed the other goat - You picked a goat, and the host reveals th…

Ah, yep, that’s right. Another way to see it is that we’re interested in the probability that your door has a goat behind it, given that you didn’t need to start over:

P(you chose goat | host didn’t choose car) = P(you chose goat, host didn’t choose car) / P(host didn’t choose car).

The numerator is 2/3 * 1/2, and the denominator is 2/3, so the ratio is indeed 1/2.

(A rejection sampling loop, where you repeatedly simulate a process until a condition holds, has the same distribution over final outcomes as the conditional distribution—so repeatedly restarting the game if the host chooses the car induces the same distribution on final results as simply conditioning on the host not choosing the car.)

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#102

Post by lachstar-x »

6gvONxR4sf7o wrote: post #13

There are a number of things I like about the monty hall problem. There's the history, the unintuitiveness, the subtle easy-to-screw-up nature of probability problems, the calculation, the sociology, and the overconfidence of wrong experts. Most of all, it's the calculation vs intuition that I like. You can do the calculation, or run simulations and prove correctness. In fact, it would be much harder to be so widely…

Can you help me understand - in my view the choice to switch doors or keep the same door is irrelevant, because even if you keep the same door you're making a choice that is now 2/3 of the right answer. If you switch or keep, you're still choosing from two doors that contain a car and a goat. The other door is no longer relevant and doesn't affect the new state at all.

It's your perspective (narrowing the choice down) that changes, not the position of the car. Like getting a run of red in roulette and thinking the next one has gotta be black.

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#103

Post by mcphage »

naikrovek wrote: post #15
saeranv wrote: post #9

I guess Monte Carlo simulations weren't well-known or easily accessible back then? It would have been the perfect way to shut up her critics.

they were. the critics didn't use them. she told them to "try it" in one or two columns before she had to explain it very slowly and methodically in a third, as I recall. I'm old. I read each of these as they arrived in the paper.

> I'm old. I read each of these as they arrived in the paper.

So back in the day, Parade magazine was a thing that people actually read, and not just ignored or recycled immediately? The past really is a foreign country.

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#104

Post by selimthegrim »

spenczar5 wrote: post #94
haberman wrote: post #76

Earlier quoted context omitted.

When you choose the first door, you are partitioning the "board" into two parts: the door you chose, and "everything else". By opening a door, Monty lets you cover 100% of the "everything else" partition using only one guess. So now you get to choose between partition 1, which covers 1/3 of the board, and partition 2, which covers 2/3 of the board.

I have never heard this formulation, and I like it quite a lot. Thanks!

It is Bayes’ Rule essentially written out (sample spaces)

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#105

Post by lachstar-x »

javajosh wrote: post #23

So this is at least in part about the "Monty Hall Problem" and why it's solution not intuitive. The article missed an important angle: when the host opens a door, he's giving you more information , which explains why it's better to switch. If you're the host, you need to know which door the car is behind to do your job 2/3 of the time, to avoid revealing it. It's this quality of unexpected information exchange that I…

Isn't he also removing relevant information as well? The third door is no longer part of the game so the information he gives you ceases to be relevant.

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#107

Post by nextlevelwizard »

macintux wrote: post #25
pmiller2 wrote: post #17

Earlier quoted context omitted.

I'm willing to forgive pretty much anyone who's fooled by something that also fooled Paul Erdos: https://www.wired.com/2014/11/monty-hall-erdos-limited-minds...

Being wrong isn’t a crime. Harshly lambasting a woman for, largely, being a woman and disagreeing with you is the problem.

Jumping to sexist assumptions is also a problem.

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#108

Post by amluto »

javajosh wrote: post #23

So this is at least in part about the "Monty Hall Problem" and why it's solution not intuitive. The article missed an important angle: when the host opens a door, he's giving you more information , which explains why it's better to switch. If you're the host, you need to know which door the car is behind to do your job 2/3 of the time, to avoid revealing it. It's this quality of unexpected information exchange that I…

I like to think about this slightly differently: at first, you bet based on no information, and you know you’re probably wrong. In advance of the host helping out, though, you have no way to put money on your initial bet being wrong. (It’s like playing roulette — the casino is not going to give you even odds betting on two dozens, which would have almost 2/3 chance of being right.)

After the host helpfully takes a door out of play, though, you can bet that you were wrong, and the payout doesn’t change. You already knew you were probably wrong, and now you can bet on this. Of course it’s a good deal. A player can probably avoid the cognitive dissonance by thinking of their initial bet as being their best guess at finding a door that doesn’t have a car behind it.

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#109

Post by ryandrake »

gonehome wrote: post #69
dllthomas wrote: post #34

Earlier quoted context omitted.

Interestingly, if the host picks randomly (and if he reveals the car, you... start over, or you get the car, or you get nothing, or ... it doesn't matter because it happens not to have happened in the time we're considering) then you are faced with a 50/50 chance.

Yeah - and at least for me this was part of the confusion. The problem is often stated incorrectly. Critically the host always removes a goat. If the person presenting the problem just says “the host opens one of the doors”, but doesn’t specify he’s always revealing a bad door then it’s not clear why that matters. The many door example makes it easy to intuit as well.

I think when stating the problem, some people assume the listener has actually watched Lets Make A Deal. You can under-specify the problem for people who have watched it, as the fact that Monty shows the worthless prize is already known.

Re: The Time Everyone “Corrected” the World’s Smartest Woman (2015)

#110

Post by dllthomas »

dunham wrote: post #84
6gvONxR4sf7o wrote: post #13

There are a number of things I like about the monty hall problem. There's the history, the unintuitiveness, the subtle easy-to-screw-up nature of probability problems, the calculation, the sociology, and the overconfidence of wrong experts. Most of all, it's the calculation vs intuition that I like. You can do the calculation, or run simulations and prove correctness. In fact, it would be much harder to be so widely…

The thing that bothers me about this problem is that (as far as I recall) it's not asserted that the host always gives the player a chance to switch. If the host sometimes gives people a choice and sometimes doesn't, based on unknown criteria, then all bets are off.

Yeah, for instance if the host only offers the switch when you've picked the car, obviously you shouldn't switch. That said, this does not seem to be the thread the respondents were pulling at...
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