Ah, the Monty Hall problem. I got that when I was 5 years old. ??? What you don't believe me? I AUGHTA... well
, I might have to accept that. But wait read on, I might convince you. I made my opening sentences deliberately arrogant, because if you can't look past your first impressions you aren't meek enough to grasp the Monty Hall problem anyway. By the end of this read you may realize I'm not as arrogant as I first came off (or maybe I am ). Besides, on this board I've already stretched my credibility to the point that its metaphor is a drumskin streched so hard it has ripped and completely receded down the sides, why not up the ante?
(If you don't know what the monty hall problem is wiki it before reading on)
First of all, I consider the monty hall to be a farse of a problem. I got the right answer at 5 because the first time I saw the game I was trying to figure out what I'd do if I was in the shoes the contestant and the answer was immediate, opposed be being posed the problem as a verbal riddle. Second of all, there is no voodu math or higher order math to this problem. After hearing the problem the typical person has a kneejerk reaction (kinda like earlier when you thought 'what a dEEk' after you read the first two sentences of this post) that locks their mind up with a reasoning equivlant to "Escape by putting the two halfs together and climb out the hole" At that point of confusion they start thinking math is so bazaar that 1+1 can equal 3, but only the einstein type knows why. It's all rubbish.
What I find even more absolutely amazing is not only--to this day--have I heard the riddle stated fairly and completely [to the person who is supposed to solve it], I've also never EVER seen anyone coherently explain away this douchebag misunderstanding of a problem, but by the end of this post I will. I've seen guru types on the web grin and gloat, assuring that the contestant should switch, and that the odds are indeed 2/3 if they do, then, maybe, offer a convoluted incoherent solution that implies a supernatural element of math. I've seen a low-functioning savant assure the switch is better, also failing to explain away the problem in simple english to those who ask. She has some forum somewhere with 100s of people posting, scratching their heads begging, wondering, thinking they might gain a superpower if they figure it out. It appears to me the savant is gloating and getting kicks, and I don't rule out that all of the above are faking it, don't even understand the problem. If they aren't going to state the riddle with with complete parameters they could at least give a hint:
"Game show hosts are nice"
I, the chosen one, at 5, walked in to the living room just as the money hall problem was starting (such a majestic entry). it was also the first time I saw the Price is Right, the first time any of us had. That is, my two brothers and two sisters were already watching it before I got there, all older roughly spaced 2 years apart. After the host opened a bad curtain of the remaining 2, my brothers and sisters were squabbling to each other whether the contestant should switch, "it doesn't matter right?" someone said. it's almost as if they had a clue, opposed to flatly declaring "it doesn't matter". I interjected "she should switch" All four of them waved me silent. "those jackasses" is a close equvlant to what I thought, and they really were. They sat infatically stumped, all open to each others words, but not to mine. Jackasses. I couped second burst, "The host is nice, right? A switch gets you two chances" They waved me away again, like I was a piece of crap. I haven't forgotten that to this day. I was basicly starting to have a mini temper tantrom at this point, pacing thinking what Sh!t life is. Then I fell to some confusion and self doubt and sort of caved like a piece of crap myself before offering my third burst: "I garentee you if she switches the chances equal or better than staying" They hushed me nonchalantly again. At this point I was so frustrated I couldn't barely remember the original chain of thought I used to 'solve' the problem. The show ended, which of course gave them no closure to the problem and no validation for me.
Later in life, after returning to a community college to pick up a few cheap credits so I could graduate, I had to take a math class (funny I don't even remember what it was, college algebra??) The teacher posed the monty hall problem to the class... he told it wrong. I didn't want to open a can of worms claiming I understood the problem, and quite frankly I wasn't prepared to spontaniously explain it. But I did raise my hand and enigmatically asked, "Can we assume Game Show Host is nice?" He sort of waved me off like I was an idiot (i AM!). To add injury (and people always do when they tell the monty hall problem, because they don't know what they are talking about) he said "No, it doesn't say anything about him being nice" I don't remember if he actualy said that, but he clearly communicated it some way.. rolling his eyes or something. I intended to laugh to myself at that point, but honestly I just felt abused again.
OK, here we go.. imagine yourself as the contestant. You chose curtain #1 (incidentally it's the wrong one but you don't know that). The host opens one of the remaining curtains (#2) to reveal some old boots. God your glad you didn't chose that one! Two curtains left, it must be a 50/50, right? WRONG. Review: you choose a losing curtain. There are two curtains left, the winning curtain and another losing one.
Now let me ask you: what kind of jacka$$ gameshow host is going to look at the two remaining curtains (one winning and one losing) and open the winning curtain and say, "this is what you could have won! now it's up to you to chose between the two remaining curtains?" That's ubsurd! The show ratings would drop, everyone would hate the host. In the other scenerio: the scenero where your initial guess is the winning curtain, the host throws out one of the two remaining losing curtains (that's truely unbiased). So ironically, if your first guess is right and you stick with it, it is an unaltered choice and you truely have a 1/3 chance of getting the winning curtain.
But if you switch, then the host's bias comes into play: he's never going to throw out the winning curtain and tell you to chose between the two losing ones. If you switch he will ALWAYS exercise his bias (which favors you) on the two remaing possibilities leaving you the winner. If you always switch the only way you can lose is if the two remaining curtains are both losing curtains. In that situation he can't use his bias algorthm, which, again, is to throw out the non-winning curtain: they are both losers. When the remaining curtains are both losers, keep in mind, that is the same scenerio in which your first guess was the winner. In that case the host's hands are tied, he can't interject his bias (that always favors you), he has to throw out one of the two losers, hence, not effecting the game at all.... leaving pure chance: 1/3 if you stay with the first guess. If you switch the host always favors you 100% , minus for the 1/3 of the time he can't (when you stay with your first guess/when both remaining curtains are losers) 3/3-1/3 = 2/3. So if you stay with your first guess you truely have 1/3 chance of winning, if you switch the host takes care of the rest: he throws out the bad curtain. The host is nice!
Another way to look at it is: 2/3 of the time the winning curtain will be amoung the two remaining curtains (conversely, 1/3 of the time the winning curtain will be your inital guess). 2/3 of the time the winning curtain is within the remaining two curtains, and the host will look at those two curtains and 100% of the time he will throw out bad curtain (he shows it to you, it's a rusty tricycle, remember?) leaving the winning curtain.
stay with initial guess: no bias...
odds of winning: 1/3, odds of losing 2/3
switch: bias... two curtains....two chances of getting the winner, and the host is so nice he pre-emptively booted out the bad curtain so fast--(before he even asked you to make a choice)--that you thought the remaining curtain only offered an unbias/nonfavored chance at winning.
Reality:
If either of the two remaining curtains is the winning curtain and you switch you have 100% of winning because the host always throws out the bad curtain (re-read that slowly about 5 times). The chances that one of the remaining two curtains is the winning curtain? 2/3 x 100% =
odds of winning: 2/3, odds of losing 1/3 (see above)
The host is probably thinking his entire career, "if you'll JUST USE YOUR MIDGET BRAIN for a moment and switch away from your inital ONE egocentric choice over to the TWO REMAINING CURTAINS, I will cover your back and take care of the rest : with two curtains left I will always throw out the losing curtain, leaving the winner. In fact, I already have: that's why you just saw five rusty bikes after I opened the curtain... I just threw out the crap for you and left the winning curtain! But I only control the situation 2/3 of the time. 2/3 is the best you can do: switch or you only have 1/3 chance of winning." At least that's what I'd be thinking every show if I were host. Most the world would have you believe only sovents can understand this problem. apparently sovents and the host--I don't think he's a sovent--but if you ask him I bet he'd say "I don't know why so few of these contestants see what I'm trying to do for them"
Ironically, not only do smart people switch, but I'm pretty darn sure stupid people tend to stay with their first guess and lose. why? let's review what the stupid person knows. Let's just say the stupid person is me so I can run you through the first person narative. The stupid person (me) distinctly remembers chosing one of the curtains. Then this crusty old guy [host] suddenly starts trying to sway me to chose one of the left over curtains instead! The crustly old guy opens one of the curtains to reveal a chicken coop with three dirty chickens, apparently in attempt to intice me to chose the other curtain. Dirty chickens?! Then the crustly old guy has the nerve to turn to me and ask if I want the other left-over curtain! The crusty old man clearly doesn't like me much or he wouldn't have showed me those chickens. If I chose the other left-over curtain the crusty old man is just going show me a bunch of junk again! But I'm not going to let him do that! My initial pick was truely a choice of free will. The crusty guy can't tamper with my inital choice. Besides, I'm a Christian American. If God wanted me to win He'd have me chose the right one the first time, and WINNERs don't have to chose from left-overs! Any American knows that! I'm sure over 1/3 of the contestants are thinking something close to that. I know, I've been around stupid people all my life; I understand their psych.
If you don't understand the problem by now I can't help you. Ok, so I didn't exactly explain the 'problem' away in only a few words. Consider this a first draft. But I think I exhaustively simplified it, using repetition at that.
So you see? the Monty hall problem isn't hard at all. The problem is that people are idiots, and these idiots that don't understand the problem, are the ones who are stating the riddle to other people, who can't solve it because the teller told it wrong!
Never have I heard someone poise this problem, then add.. "by the way, if the contestant switches the host always throws out losing curtain and lets the winning one remain, and if both curtains are losers he just throws either one" Gee, if that vital part of the problem was included do you think people might have a better chance at figuring out the answer? Poised as a logic question it's not considered correct to assume the host is nice, but it's implicit if you are refering to the actual show (no host is going to show you what's behind the winning curtain then ask you to chose one of the remaining losing curtains)
To add more insult to injury, about 10 years ago, I asked my brother, the smart successful one, the one that's rich, in manhatten beach. Yet, he only seems to be employed about 5% of the time. I hate that guy. Anyway I ask him, remember that time when we were all little at the first house and we saw Let's Make A Deal and we were all trying to figure out which curtain to to pick?
His Answer: no
the abuse never stops when your right.
One thing did give mea little artificial self doubt through the years: I found it a bit odd that all five of us kids saw Let's Make A Deal for the first time at the same time. Today I solved that one though. According to wiki, The show was remade in 1972 .... THE YEAR I WAS FIVE (there was an earlier version in the 50s it said).
And for those of you that who are trying to decide whether I'm a super genious or a typical arrogant liar.... here's news for you, it doesn't have to be either:
Children know that Gameshow Hosts are Nice.
http://www.youtube.com/watch?v=lfuUxFtvXDU&NR=1
more info about "it doesn't have to be either":
http://www.giftedchildtips.com/video.php/1039
So I'm not so arrogant after all, or am I?