Variable Change - 'The Monty Hall Problem' - 21: The Movie

Kevin Spacey and the character, Ben Campbell, from the movie "21" explains the Monty Hall "Game Show Host" problem. This is a very good description of variable change

[ADRIANCASTRO-jy]

Amazing representation! Great Spacey...

[TedManney]

"This is a very good description of variable change" says the description. Yeah, because "variable change" isn't a real thing, it was made up by the screenwriters of this movie as a technical-sounding woo-woo term to stand in for a coherent explanation of the problem.

The swapping advantage depends on the host always eliminating a goat no matter what. If he's doing anything else, the swapping advantage doesn't necessarily follow. And if he's only offering the option to swap conditionally upon the player picking the car off the bat (the scenario Kevin Spacey suggests), then swapping is the *worse conceivable strategy* because it will win a goat 100% of the time.

When the student says he doesn't care about that and is just "playing the probabilities", what he's really saying is that he doesn't understand what information feeds into the swapping advantage and what information does not.

[edwinrios]

I had to think hard about this short video to understand the concept of variable change. The premise of winning the car after switching from door 1 to door 2 doesn't seem intuitive, or even logical, being that you only have a 50% chance after door 3 was revealed as a goat. However, when door 3 was reveled as a goat, you already won 1 selection. You didn't get the car, but you did win a selection of, lets say, eliminating a door that had a goat. Now you are offered a second selection. Since you already eliminated 1 door with a goat, does that mean that over time you will hold a 100% record for eliminating all doors with goats? Maybe not, and since you were able to eliminate the first one already, there is a chance that the second option may not allow you to eliminate the second goat with the same choice. So you have to think that this could also be defined as a game of eliminating goats. This does not guarantee that the car will be behind door 2, but given 2 opportunities to eliminate goats, chances are you can't eliminate them by maintaining the same choice always. So you are actually working on the premise of being 50% right and 50% wrong with the same choice. That is where the logic is, far from the emotion of believing that given a second opportunity, you will always be right with your first choice.

[jszlauko]

If you ALWAYS switch doors when given the opportunity, you WILL win 2/3 the time. If you ALWAYS stay with the door you originally selected, you will win only 1/3 the time. Thus, by always switching, you double your chances of winning. It's amazing the people I run across who just cannot understand this. I think if you want to get mathematical about it, it has to do with something called conditional probability. However, the answer is obvious if one simply sees it this way...what are the chances that your original guess for the car is correct? OK, we know it's 1 out of 3, or 33%. So, stay with your original choice, you'll win 1 out of 3 times. Now, what is the chance that your original choice was the goat? It's 2 out of 3 chance, since 2 of the 3 choices are goats. If you are going to switch every time, it means that whenever your original choice was a goat, you'll win! We just saw that you have a 2 out of 3 chance of selecting the goat with your first guess, thus if your switch, you will get the car 2 out of 3 times.

[stevew]

1 and 2 are indistinguishable after the 3 is revealed. 1 and 2 and 3 all started at 1/3 but after 3 is revealed, BOTH 1 and 2 changed from 1/3 to 1/2. There is NO MORE probability of winning by switching from 1 to 2. It became a new problem altogether when the game show host eliminated 3 and asked you at that point whether 1 or 2 is preferable. At that point, 1 and 2 became indifferent. They are both equally preferable at 50%. The error in the movie makes for an interesting script but it doesn't make it right.

The movie also implies card counting is illegal for which casinos can drag a practitioner of it into the backroom and beating him up for cheating, both are also wrong. Card counting is legal, but casinos do not condone it and they are a private business so they have the right to kick people out (not to beat people up in back rooms) for card counting. They can't arrest people for card counting, but if the better refuses to leave after being warned to leave he can be arrested for trespassing but he can't be charged for cheating.

[jimmyma]

sure the odds are better for him with only 2 doors remaining, but what IF the car IS behind door number 1?   he looses a car..

[nawinaswinsakthivelan]

how is it 66%...for all know it should 50% right ?