Finally - A Monty Hall Problem Explanation that makes sense, even to me!

[Arnold]

Found the Monty Hall thread very interesting after my flatmate referred me to the thread on this site about it.

Never really got it myself (lots of arguments) until I read an explanation that apparently has had great success in convincing people, even some maths academics, that it is actually a 33/67 proposition:

Source: Understanding the Monty Hall Problem | BetterExplained

One explanation was as follows:

Perhaps an easier way to get people to understand the problem is to disprove the 50/50 theory rather than attempt to prove the 33/67 theory.

A simple way might be to ask the proponent of 50/50:

"If you ran a game, such as on "Let's Make a Deal", where contestants are given a one-in-three chance of picking a prize but in this case they are not allowed to switch, on average, how many contestants would you expect to win the car based on their first random pick from three doors?"

Hard to imagine that anyone, even a 50/50 proponent, could answer anything other than "33" or "About 33".

You would agree with this and perhaps say:

"Then how can each of the two remaining doors in the Monty Hall problem offer a 50% chance of winning?

If this were so, 50 contestants would each win a car if no contestant switched and, ipso facto, 50 contestants would each win a car if all contestants switched."

"However, as we've agreed, only 33 contestants on average can win a car for every 100 contestants when no one switches. So how can the number of contestants who pick the correct door in a one in three guess suddenly increase from 33 to 50?"

Presuming the dropping of the naysayer's jaw will prevent them from attempting to explain this conundrum (and we hope they won't persist in claiming that the order in which the unopened doors are opened somehow increases the odds of winning...) one might say:

"If 33 win when no contestant switches, 67% must win when everyone switches (because the 33 who won when no one switched will now lose and the 67 who lost when no one switched will now win)."

So not only can it not be a 50/50 proposition, it must be a 33/67 proposition as the number of contestants who win can be expected to double when all contestants switch doors.

Even the proponents of 50/50 would seem to have to agree with the proposition.

I have read many explanations of the Monty Hall problem but never one that focuses on why "50/50" cannot be right rather than why "33/67" is right.

It would seem that if you can get the 50/50 folks to agree that you do not have a 50% chance of winning if you do not switch, then that would seem to leave 33/67 as the only plausible theory.

If the chances of winning a 1-in-3 guessing competition cannot be 50%, they must be 33% for the first door picked (i.e. 1/n, where "n" is the number of options from which it was chosen), and 67% for the other (n-1/n), as the sum of all probability must add up to 1 or 100%.

The larger the value of "n", the more advantageous it is to switch. Many people have shown this when they point out the analogous scenario of a game comprising 100 or one million then, usually in desperation, one billion or perhaps, a Googol doors to show that the probability of the first door being a winner very much depends upon the number of doors from which it is chosen, as with any random guess, not the number of options remaining after all of but one of the duds have been eliminated.

: : :

I thought that explanation was pretty hard to argue against.

[ParaBul]

I've never encountered this problem before, and I admit my first thought was "50:50".

After reading the above (and a few other discussions from the linked page) I find it easiest to get my head around it this way:

After choosing a door, there is a 2 in 3 chance that the car is behind one of the non-chosen doors. By selectively (NOT randomly) removing one of the non-chosen doors there is still a 2 in 3 chance that the car is within that set - which is now only one door.

I think the key point is that the door that Monty opens (or eliminates from the set) is not chosen randomly. That's why the odds become so much better.

Edit: this is messing with my mind now.

For a new player to now enter and be presented with the two remaining doors, they have a 50:50 chance of randomly picking the car. For me, the odds are different because I have different information, so my choice is biassed - I'm not randomly picking any more.

I remember reading something similar in a novel many years ago. Two men were taking it in turns to call the outcome of a coin flip, and one was having more success than the other. The more successful man had eight coins in his pocket; five were manufactured in one decade, and the other three were manufactured in a different decade. If he pulled one of the five coins, he would call heads; otherwise, he called tails. The coin toss was completely random, but his call wasn't.

Of course, in a novel this works. In real life? I have no idea.

Edit 2: tried playing with the coin-toss game in Excel, using the random number generator. After more than 87000 coin tosses, the non-random call was up by 142 wins. I don't think that's a statistically significant result, so I'd say it doesn't work.

I'll do some drinking now - that should stop me thinking about this.

[team_v]

This has been done to death, no one cares.
No point in making a new thread about it.

[elephino]

The answer is, you'll never win the car because the game is rigged...oh, no, that's 3 card Monty

[ParaBul]

This has been done to death, no one cares.
No point in making a new thread about it.

Done to death where? I read a lot of crap on the internet, have done for 8 or 9 years, and it's the first time I've seen it. I found it interesting, and appreciate his post.

Obviously, I care, and you have declared me to be no one. What makes you so special?

[team_v]

Done to death where? I read a lot of crap on the internet, have done for 8 or 9 years, and it's the first time I've seen it. I found it interesting, and appreciate his post.

Obviously, I care, and you have declared me to be no one. What makes you so special?

Because this is a car forum and the question was posed a year or so ago and started a whole bunch of flaming so it got canned and people were blocked.

[ParaBul]

Because this is a car forum and the question was posed a year or so ago and started a whole bunch of flaming so it got canned and people were blocked.

Forum: General Discussion

This is where you can have your say on anything that doesn't fit in the other sections

Anything.

This fits.

Flaming? Yes, that's an unfortunate part of discussion forums, which is why they have moderators. So far, the closest this topic has come to flaming was instigated by you.

Here's a suggestion, and it's quite a common one in discussion forums: If you don't like the topic, don't read it.

Is your life reallly that poor that you need to stop others from enjoying something that you don't?

[The_Hawk]

Guys,

team_v is correct, it was done to death on here and was canned because it ended up in conflict, many deleted posts, many PM's and some enforced holidays away from the forums. Not something anyone wants to see happen again.

This topic, and a few others I could think of, tend to end with people fighting for no apparent reason. Essentially both sides of the debate adamant THEY were correct and the other side just couldn't see the truth.

Taken from the Mythbusters Wiki:

The [Monty Hall] problem is presented, discussed, and tested in the television show MythBusters on 23 November 2011. This paradox was not only tested to see if there was an advantage to switching vs. sticking (which, in a repeated sample of 49 "tests", showed a significant advantage to switching), but they also tested the behavior of "contestants" presented with the same situation. All 20 of the common "contestants" tested chose to stay with their original choice.

Hunt up the episode as it's a decent watch, although pretty much covers what is in the first post here and supports the "you should switch camp".

I'm going to leave this thread open for now, but if it runs off the rails it's getting locked. I also don't want to see 20 responses saying how it wont if people just play nice or move on if they don't like it etc etc. It's only half a step away from waving your finger in your brothers face while saying "not touching, not touching" then yelling for mum when he smacks you for being a d--k.

[Arnold]

Thanks, Aaron, I did not know about the Mythbusters episode but could only find a French version on youtube. But these guys are no dummies, to put it mildly, so perhaps the mere mention of their conclusion will satisfy anyone still harbouring any doubts about whether the problem is a 33/67 proposition.

I did find a video with an explanation here (Mythbusters: The Monty Hall Problem) which is really clear and easy to understand if anyone is still harbouring any doubts.

The clip is also VWWA-compliant as the car featured is, in fact, a VW! What are the odds the video I found would feature a newish VW?

Does anyone know what type of VW it is?

[Arnold]

Because this is a car forum and the question was posed a year or so ago and started a whole bunch of flaming so it got canned and people were blocked.

I just had another look at the original thread on this topic and could not see any flaming.

I did note however that the more vitriolic and impassioned posts (and use of upper case as in "WRONG") were from those who believed the MH Problem led to a 50/50 proposition with no benefit to switching from the original door chosen to the only remaining door.

While you were a strong proponent of the 50/50 view, team_v, your advocacy was always courteous (unlike one user "Rocket36" who went from saying how "cool" it was that people could have a civilised discussion to saying people who did not understand that it was 50/50 should go back to primary school!).

So credit due for your civility, team_v.

While there was no apparent "flaming" (i.e. personal abuse), some people, 50/50 proponents it seems, starting with "Buller_Scott", posted a lot of off-topic messages, seemingly to divert the thread from its purpose.

There was a follow-up thread which also attracted attention from people who were uninterested or did not care about it and who made comments such as:

What is this puss?? Who gives two sh*ts about it?? It was closed for a reason.
Why don't we leave the crap discussions in the Mk6 DSG thread...

The apparent reason the thread was closed was that it was going "round in circles".

When the thread was closed, some people laboured under the misapprehension that MH was a 50/50 proposition, which is a shame, because it seems the penny was starting to drop for more and more people.

Anyway, this is great section in a terrific forum. Interesting threads are allowed to flourish and are not closed due to protests from people who are either uninterested or uncomprehending of the content or want to control what other people discuss (as happens in China).