Exactly right on all counts. By the way, the co-founder of Mensa calls puzzle-solving “mental masturbation.” It’s enjoyable enough but it doesn’t actually accomplish anything in the end.
She is often wrong and will not admit it. I remember some years ago reading something very wrong about the length of day but I cannot find it now.
Cecil may ocassionally be mistaken but does continue with the issue until the truth is clear which is the whole point. Marylin just ignores people who point out her mistakes so it is not about truth, it is about her.
Chronos wrote:
Actually, the columns are written by a computer known as CECILAC.
After one key is drawn, and you know it doesn’t work, the odds of the next key working are indeed 1:6.
I didn’t see the original column, but what I think the problem asks is that what is the probability that, without knowing whether the first key will work, the second key will be successful. Once you have knowlege not granted by the original question, the situation (and thus the answer) changes.
Without knowing whether or not any of the other keys work, the probability any one key in a set of n keys will work is 1/n. After you know a key doesn’t work, you now have a set of n-1 keys and the probability (really a measure of your ignorance, which has now lessened) of any one remaining key is 1/(n-1).
Not Marilyn Vos Savant, the manhole question, and the Monty Hall problem again. NO NO NO NO NO!!!
Cogitoergosum originally asked:
Several people have partially addressed this. First, Marilyn was scored with her IQ of 225 as a child using the antiquated (outdated, no longer valid method) of MA/CA x 100. That method is to measure “mental age”, divide by “calendar age”, and make a percentage. As noted, that problem was especially unreliable with children. It was replaced with a statistical method of comparing IQ. The current version does not allow IQ’s over about 180, as they are statistically non-meaningful. Marilyn retains the Guinness record for highest IQ because she rated that under the old system, and then the system was changed so that no one can ever score that high again. Technically she is entitled to the rank, but it really isn’t an accurate title.
Also it was noted that someone who takes a lot of IQ tests can develop a familiarity with the questions asked and thus perform better on later tests. I bought an IQ test book a while back. The first test I scored about 120, but on the next two or three I ranked higher - up to 132 - because I figured out the methods to some of the puzzles. Once I saw the puzzle I recognized the types of tricks to them and were easier to solve than the first time, not knowing what the trick was going to be.
As for comparing IQ’s, or what they measure, that’s not something that necessarily makes sense. There has been lots of discussion on this (not necessarily here), including a system that describes something like 8 different types of IQ. Differences in muscical perception and understanding vs mathematical comprehension vs. language interpretation vs spatial relations. The point being that different people are better at different things. Typically standard IQ tests do word and math games and puzzles, with the occasional spatial relationship. Of course ideally you would want the puzzles to be beyond the person’s level of study, so that they have to use creativity to solve it, not memory of something already learned. Also, because of the statistical nature of the results, comparing between tests is less meaningful.
Regarding the manhole cover question, of course it’s the lip that keeps it from falling in. Duh. She says manhole covers are round because manholes are round. But that begs the question “Why are manholes round?” And that is because it is the best (simplest) shape such that the lip can be small and still keep the cover from falling in. Sure you could use a square cover and a square hole and a big lip, but then you have to make the hole even bigger to have enough opening for a person to fit through. The whole point (get it?) is to minimize the total hole/cover size and allow maximum opening for a person/equipment to fit through. Thus the circular shape.
I think you’ve solved the key question. Looking at it from the beginning, the probability it will be key 7 is the same as it being key 1. However, if you start trying keys and discarding them, at that point you are changing your information and change the probability.
Regarding the Monty Hall problem with the doors and the goats, I tried an archive search for the half dozen threads where this has been beat to death before, but curiously when I used “Monty Hall Problem” I kept getting a list of threads on everything from basketweaving to the shoe size of someone’s ex-girlfriend. (Okay, maybe not those topics, but just about as irrelevant.)
The column link was provided, and Cecil covers it.
SingleDad is right, Marilyn is a dilettante. She phrases her responses poorly - specifically, but poorly. By that I mean reasonably intelligent people are thrown off by her unstated implicit assumptions. And regardless of whether she made up the questions herself or is responding to letters from readers, the questions she answers are fluff. Then again, some of Cecil’s columns are less than earth-shatteringly important. But at least he tackles the burning questions nobody else will.
How do lava lamps work?
Can hair turn white overnight from fright?
Who gets the most pleasure from sex, the man or the woman?
What’s up with vacuum cleaner wounds to the penis?
In Steve Miller’s “The Joker,” what is “the pompatus of love”?
Is it possible to recall past lives through hypnosis?
What’s the story with female circumcision?
Why is, um, fecal matter brown?
[/blanket post]
(Yea for preview!)
[southern belle drawl on]
Marilyn vos Savant? Oh, we hate her!
[southern belle drawl off]
Peace.
I just read the Cecil column referenced in the post which followed this one. I think Cecil got it right (the second time), but the point got a little diluted with arguments about semantics of the problem statement and what Monty Hall likes to do.
I love this little problem, and will admit that I didn’t buy it at first, either. Stating it my way: 
It’s a game show. There are 3 doors: behind one is the prize; behind the other two is nothing. The host knows going in which door hides the prize.
The contestant selects a door, but it is not opened yet. The host then opens one of the other doors, one which he knows will reveal an empty room. The contestant is then given the option of either opening his initial selection, or switching and opening the (one) other unopened door.
Is it statistically in his favor to switch, not switch, or does it matter?
At first blush, it appears pretty obvious: it doesn’t matter.
This is incorrect. It is in the contestant’s favor to switch, and the best explanation I’ve come across for why this is so points out this fact:
If he chooses to switch, he will win if his initial guess was wrong.
There is a 1/3 chance that his initial guess was right, and a 2/3 chance that it was wrong. Switching, then, will result in a win two times out of three. Not switching will win only if the initial guess was right - this happens only 1/3 of the time.
Brad, this has been beaten to death and you are mistaken. It is so easy to see (I will not even say “prove”) that I will not even bother. Analysis should suffice but just do all the possible combinations and see the results.
Oh, boy…
I’m sorry sailor, but I disagree. You contend that the answer is so obvious that it needs no proof. I contend that the “obvious” answer turns out to be incorrect upon more careful analysis.
I’ve attempted to offer a (very brief) analysis. You’re dismissing it without rebuttal, saying merely that I should “do all the possible combinations.”
Very well. The problem is as I stated in my first post.
Without loss of generality, suppose that the prize is behind door #1. This divides the number of things we must consider by three.
The only remaining variables are which door the contestant selects, which door the host opens, and whether or not the contestant switches. The host cannot open door #1, since it contains the prize. I present the following table of combinations:
Contestant choice – Host Opens – Switch? – Result
1 2 Yes Loss
1 2 No Win
1 3 Yes Loss
1 3 No Win
2 3 Yes Win
2 3 No Loss
3 2 Yes Win
3 2 No Loss
Eight possibilities, four of which result in wins. Two of those wins come on switches, and two come on stays. Seems to imply that it’s 50/50, doesn’t it?
It doesn’t, for this reason: “counting the combinations” in this simple fashion only works if they’re all equally likely, and in this case they aren’t. Notice that four of the above combinations have the contestant picking door #1 with his first guess. Really? What clairvoyance does he have that allows him to pick correctly on his first guess half the time, rather than 1/3?
You have to halve the weight of the Contestant-picks-#1 combinations in order to make things work out. When you do this, you obtain the 2/3 bias towards switching that I’ve been harping about.
This is a very easy-to-make flaw in analyses like this. You have to be careful doing it this way! That’s why I’d really like to see somebody show why this statement of the results is incorrect:
1/3 of the time, the contestant’s initial guess will be correct. The host will open one of the other two doors (neither contains the prize, so it doesn’t matter). If the contestant “stays” he wins, and if he switches, he loses.
2/3 of the time, the contestant’s initial guess will be incorrect. The winning door will be one of the unchosen two. The host is forbidden by rule from opening the winning door, so he will open the unselected empty room. If the contestant stays he loses, and if he switches he wins.
I don’t know how much more simply I can state my case. Write a computer program to simulate it if you don’t believe me. I did it, and it’s true.
I’m just imagining what a great slogan: “I’m the 500th smartest person in the world!” would be.
sailor and brad_d, here’s another way to look at the problem: Suppose that you pick a door, and the host gives you the option to keep door number one, or take both number two and number three. This is essentially what is being offered here. You want to switch.
As dicussed in other threads and mentioned by Cecil, this does still require some assumptions about the game.
Brad, I am sorry I overspoke myself as you are right. I lost a good opportunity to shut up and I should have checked it out before saying anything. Sorry I made you do it rather than doing it myself.
Looking at the 8 different possibilities you mention it is easy to see the probability of any of the first four is 1/12 and the probability of any of the last four is 1/6. Now we can add up all the probabilities:
Switch and win: 1/6 + 1/6 = 1/3
Switch and lose: 1/12 + 1/12 = 1/6
Keep and win: 1/12 + 1/12 = 1/6
Keep and lose: 1/6 + 1/6 = 1/3
In other words, if you switch you win 2/3 of the time, if you keep you win 1/3 of the time.
I’ll try not to do it again…
OK, I think I finally get how to explain this Monty Hall condumdum. Shroedingers cat is either behind door number one, or he isn’t. His state is unknown. As soon as Monty opens door number two, the cat behind door number one is dead and the cat behind door number three is in a state of indeterminancy, mixed with anxiety. However, S’s cat is indeterminate behind all three doors until one of the doors is opened.
Q.E.D., IMHO
No apology necessary, sailor. I’ve done the same thing more times than I like to admit.
Chronos, you’re right - it does require some assumptions about the game, as I gather that this isn’t quite the way Monty Hall played it. I like your way of thinking about the solution, too. It hadn’t occurred to me.
It hadn’t occured to me, either… The board has run this one into the ground plenty of times, I saw that explanation somewhere else before, don’t know who suggested it.
The real reason Ms Vos Savant is so inferior to Cecil is that her questions come from people who read Parade while his come from readers of the Reader and this web site.
Hey, I’m a newbie. I have to keep sucking up.
That’s pretty close to what I say. Now and then I pick up Parade as I’m discarding the dreck from the Sunday paper ad inserts. I see her column and find myself wondering, “Now, if I set myself up as ‘Ms. Smart Person’, why would I bother publishing replies to dumb questions that don’t test intelligence?” A lot of them seem to be more up an advice columnist’s alley-- why would she bother with these?
$0.02
brad_d, your analysis is lacking something critical. You neglected to list all the times when the contestant picks the wrong door, and Monty doesn’t offer to switch but instead just opens the one he picked. The problem doesn’t state that Monty must offer the switch.
Now re-do your analysis, first finding the optimal strategy for whether Monty should even offer the switch, then determining the contestant’s strategy for sticking/switching. You’ll find that it’s 50/50.
CurtC, I’m certain you’re right. It would be an interesting analysis, and I’ll try to do it sometime - sounds like fun. I guess the goal (from Monty’s point of view) is to come up with a strategy that will make it 50/50.
You’re correct that the game, as I defined it for my own analysis, omits that critical detail of Monty Hall’s game. I sort of did that on purpose: even without it, it’s an interesting problem that often surprises people - if they can be convinced to believe the correct answer.
So…yeah, I kind of ducked out of something, and I apologize for any confusion that may have caused. Until I read Cecil’s column a couple of days ago, I hadn’t realized that Monty Hall or anybody else actually had such a game for real. The “simplified” version had been presented to me as a sort of “hey, get this” intellectual exercise in college.
So I stumbled onto MvS’s website and saw her brain e-teaser of the week. The question was, what did Alexander Graham Bell suggest as a telephone greeting? The answer she gave was “For some reason, Bell liked ‘Hoy, hoy.’”
Of course, in reality, Bell liked “Ahoy, ahoy” for the obvious reason that it was already in use as a nautical hailing call.
what I find a little weird about MvS’s errors is that sometimes I have to wonder if she’s even trying. I think this e-teaser must have originated in her saying to herself, yeah, wait a second- didn’t Bell think people should answer the phone with “Hoy” or something? A simple trip to the library would have avoided (another) embarassing error in front of thousands of readers, so why didn’t she do it?
I also found the Bad Astronomy page which discusses MvS’s wrong answer to the elevator problem (“If you are in a falling elevator, can you save yourself by jumping right before you hit the ground?”) Now, this is a question I think every science columnist must get all the time, and I also suspect that most people who read a lot of science columns have already seen the answer- it’s like the perennial “Why is the sky blue?” So when Marilyn gets it wrong, I find that to be a little… weird.
-Ben