In a magazine article in Parade in 1990 ( http://marilynvossavant.com/game-show-problem/ ), Marilyn vos Savant instructed her readers that if ever offered, they should always choose to switch doors. vos Savant was correct in her analysis, but that did not stop thousands of readers, some even very highly educated, from writing in to voice their opinion that vos Savant was wrong in her analysis. All of the respondents, including some highly skilled mathematicians, used logical fallacies in their attempts to prove vos Savant wrong.
It was only after being shown proofs and computer simulations that supported vos Savant's conjecture that the mistaken arguers began to back down. The large amount of false accusations, even by learned scholars, should show that all information has some level of uncertainty. It would be unwise to take a argument or fact at face value without looking into it yourself.
Image from http://stevenlarose.blogspot.com
Consulted source: http://en.wikipedia.org/wiki/Monty_Hall_problem
How did the people arguing against Savant use fallacies? Are all the people who immediately didn't believe her guilty of committing a fallacy? Couldn't it have still be smart for them to be skeptical even though they were wrong and as a result would have made no mistake in reasoning. They must have made some sort of a mistake somewhere because they did end up with a false belief but it seems that the requirement for committing a fallacy is stronger than that.
ReplyDeleteWhy did all of her readers want to prove her wrong in the first place using logical fallacies? Did they get pleasure out of proving someone wrong? I find it astonishing that highly skilled mathematicians did not understand the logic behind the Monty Hall problem and that they had to use logical fallacies to try and prove her wrong. An important aspect of this problem is that it induces a fake sense of reverse psychology. During a game show someone who is not very math averse would probably stick with their chosen door because they may feel that the game show host is trying to trick them into choosing the other door. But for highly skilled mathematicians to have to resort to using an error in reasoning to prove a point surprises me.
ReplyDeletehttp://montyhallproblem.com/
http://www.logicalfallacies.info/
I'm not too familiar with the notion that the Monty Hall Problem is a "classic" example of logical fallacy; though the solution is not intuitive and (as mentioned) difficult for even experts to accept, were the people disagreeing with vos Savant using logical fallacies in their arguments against her? (i.e. attacked her character). Otherwise, in my opinion, just disagreeing with vos Savant does not in itself make one commit a logical fallacy despite her being correct.
ReplyDeleteCody and Gordon, great comments, here is some additional information that I probably should have included in the original post regarding specific responses. At the website http://www.oocities.org/inkwire/workshop/intro/montyh3.htm you can find 25 different responses to the original article in Parade with logical fallacies in them. Here are a few in specific...
ReplyDelete"Dear Marilyn:
Since you seem to enjoy coming straight to the point, I'll do the same.... you blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error and in the future being more careful.
Robert Sachs, Ph.D.
George Mason University"
"Imagine three runners in the 100 meter dash. The runners are so evenly matched that the odds of any given runner winning are completely random. I am asked to guess the winner and select runner number 1. After making my choice, runner number 3 pulls a hamstring and cannot compete. Based on your [Marilyn's] answer to the game show problem, I would conclude that the unfortunate injury to runner number 3 has somehow inexplicably made runner number 2 the clear favorite over my original choice. By way of the above example, I hope it is now clear that your analysis of the game show problem is flawed.
Jeffrey Hoyt
Department of Mechanical Engineering
Washington State University."
"There is such a thing as female logic. Since Marilyn is female, and thinks it's logical to switch, then that's the right answer."
The third example is the most blatant of the logical fallacies. Obviously, a person's gender has nothing to do with whether or not they are right or wrong. It's not surprising that the person who made this comment chose not to identify themselves.
However, the other two responses come from highly skilled mathematicians who chose to either ignore some statistical laws or use incorrect analogies.
After reading this post and its comments, I was very curious what vos Savant's proof was in favor of always switching doors given the chance, as no one has mentioned it yet. After researching and trying to wrap my head around it, I have gathered the following: The guest originally has a 1/3 chance in picking the correct door. "Switching" would only not work if the guest chose the prize in the first place, which again is a 1/3 chance. Switching would result in the compliment probably to that, which is 2/3. Therefore, switching doors will always have a 2/3 probably in choosing the prize, while not switching results in a 1/3 chance.
ReplyDeletePersonally, after thinking about it, this logic makes sense to me. However, and in response to Gordon Wu's question, the people disagreeing with vos Savant, which included the general public and university professors, did not commit logical fallacies themselves. Nearly 100% who ran one of vos Savant's follow up simulations changed their minds and agreed, which included 71% of university professors. vos Savant also later stated that some confusion was caused by some readers not realizing that the host must always reveal an empty door, and does not in fact reveal a random door. This resulted in many other readers changing their answer as well.
Thanks for the explanation Brett. The way of describing the situation that really made the light click for me when I first researched this topic was roughly as follows. Instead of imagining a scenario with three doors, imagine a scenario with 100 doors. 99 of the doors are trash and 1 is the prize. When you first pick a door, you have a 1/100 chance of getting the prize. Now the host eliminates 98 of the wrong doors. You've always had a 1/100 chance of picking the correct door on the original guess, so why would the host automatically removing 98 incorrect doors AFTER your guess change it? That's just what helped me.
ReplyDeleteI did the Monty Hall Problem back in high school in my statistics class. At first thought and I think most people and mathematicians thought that once a door is removed it becomes 50/50. That is the easy conclusion. I believe that many people had never thought about this problem before and when they did their gut feeling told them that it should be 50/50. Now I think most people trust their gut so that when my gut tells me something I will argue that point until someone can make me believe otherwise. I think that at the beginning because of the issues that vos Savant had explaining what happened and because of the gut feeling many people have that many of those people pulled fallacy after fallacy to prove vos Savant wrong. Kyle's explanation with the 100 doors makes a lot of sense and it makes it much easier to understand the complexity that this problem brings. I also think that when you only think of it within 3 doors it makes it much harder to deduce the real situation that is going on.
ReplyDeleteGreat post, Kyle. "Lets Make a Deal" is a classic game show that offered the option to switch prizes, but I wonder if the Monty Hall Problem also applies to new game shows. "Deal or No Deal" also uses the switch option, often to the detriment of the contestant, but all of the prizes aren't junk. Having a spectrum of prizes (good, bad and okay) seems to radically change the incentive structure of the game. "Deal or No Deal" contestants also fall into the Ad Hominem fallacy when they deny the game show buyout options, believing the shows designers are always low-balling them, or are not trustworthy enough to offer a good deal. In act, I'm pretty sure the show uses a strict mathematical formula when offering the deals to the players, meaning the deals are probably always fair.
ReplyDelete