Author Topic: SGU 5x5 #99  (Read 2124 times)

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Offline Mike

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SGU 5x5 #99
« on: January 06, 2011, 09:11:39 PM »
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Offline Technogeek

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Re: SGU 5x5 #99
« Reply #1 on: January 08, 2011, 03:56:44 AM »
I'm reminded of one book I read as a kid. I can't remember the title, but it was part of one of those "groups of kids acting as detectives" series that I imagine were popular at some point in the past.

Anyway, there was actually what is in retrospect a surprisingly skeptical look at this subject. Said group of kids had a handwriting sample related to what I think was some sort of insurance fraud scam, and brought it to the kid genius member of the group in hopes of figuring out who it belonged to. Under the impression that graphology (though never referred to by that name) was legit, they asked said kid genius to figure out the personality of who wrote it so that they could track down the culprit.

What the kid genius did was pretty clever. He took samples of handwriting from the other members of the group under the pretense of a demonstration, and swiftly matched each sample (which had been written with his back turned) to the correct group member based off personality traits revealed from the nature of the handwriting -- immediately after which, he explained that what he had actually done was just a bit of hot reading (mostly going off of what was written and not in what hand), and that real handwriting analysis was basically graphonomy.

I really wish I could remember the name of that book. It was a fun read when I was young.

Offline Vincegamer

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Re: SGU 5x5 #99
« Reply #2 on: January 08, 2011, 10:18:30 AM »
Sounds like Encyclopedia Brown. 
I don't recognize that particular story, but those books always took the form of the kids encountering some mystery or a scam perpetrated by the local bully and taking it to Brown the boy genius who solved the problem.
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Offline Technogeek

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Re: SGU 5x5 #99
« Reply #3 on: January 08, 2011, 11:59:15 AM »
No, it wasn't one of the Encyclopedia Brown books. That much I remember.

Offline clavicorn

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Re: SGU 5x5 #99
« Reply #4 on: January 08, 2011, 09:25:44 PM »
Dear Evan,

Goethe = "gertuh," not "gertay."

Offline rmcbride

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Re: SGU 5x5 #99
« Reply #5 on: January 08, 2011, 10:19:09 PM »
Heard of this in elementary school.  Wasn't called Graphology though.
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Offline tarrou

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Re: SGU 5x5 #99
« Reply #6 on: January 12, 2011, 10:57:19 PM »

(Please only bother reading this if math is fun for you.)

I have a small clarification to this episode's discussion about Randi's experiment with the graphologist.  To remind the reader, there were five women who provided handwriting samples to a graphologist.  He was then expected to determine the vocation of each woman based on these samples.  He got one of the five vocations correct.

Rebecca then said that this was exactly what we would expect due to chance.  (I watched the video of Randi on YouTube and he said something similar.)  This is right in a technical sense, but I think it's a bit misleading, especially with the use of the word "exactly".

This is a big assumption, but I'm guessing Rebecca and others were thinking of this experiment as follows, since it's the common scenario.  For each woman, there were five professions from which the graphologist could choose.  Thus, if he just guessed for a particular woman, he would have a 1/5 chance of getting it write.  If he then guessed for the five women, he'd likely get one right, seemingly corresponding to what actually happened.

However, this does not really describe the situation.  That's because after he guesses for the first woman, he only has four professions to choose from for the second woman.  (The experiment was designed in such a way that he knew two women didn't have the same profession.)  Continuing in this manner, he would only have three professions to choose from for the third woman, etc.

Think of the same experiment with just two women.  If the graphologist guesses the correctly for one women, he automatically guesses correctly for the second woman, because there's only one profession left to choose from.  If he guesses incorrectly for the one, he guesses incorrectly for the other.  Thus, half the time he gets both professions correct and half the time he gets neither profession correct.  He never gets just one profession correct.  This is interesting because in the interpretation I attribute (possibly incorrectly) to Rebecca, one would guess he was expected to get one correct by chance.

If anyone is interested, I calculated that, assuming the graphologist just guessed, we'd get the following probabilities:

0 correct -- 36.7% 
1 correct -- 37.5%
2 correct -- 16.7%
3 correct -- 8.3%
4 correct -- 0%
5 correct -- 0.8%

So yeah, he'll most likely get 1 correct, so Rebecca's statement is technically correct, but 1 correct is just barely more likely than 0 correct.  (The graphologist did kind of well, considering.)   However, when hearing the experiment, it occurred to me that this was not a normal situation and I thought it'd be fun to look at.  It's cool, for example, that getting 4 correct is impossible.

I know; of course the first time I decide to post to the forums it's a criticism, and that makes me a bit of a jerk.  So I'm sorry.  Thank you for the podcasts; I listen quite regularly.

Offline Chew

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Re: SGU 5x5 #99
« Reply #7 on: January 12, 2011, 11:34:10 PM »
Math is fun! Welcome to the forum. If you're planning on sticking around for awhile you can introduce yourself in our Introduce Yourself Here thread.

Is there a quick algorithm for computing the odds in situations like this? I've seen psychics tested like this before and I'd love to be able to calculate the probability distribution because the host always says the simple 1/n probability which, as you pointed out, is wrong.
« Last Edit: January 12, 2011, 11:37:02 PM by Chew »
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Offline tarrou

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Re: SGU 5x5 #99
« Reply #8 on: January 13, 2011, 09:49:54 PM »
Hi Chew,

I'll have to go to the Introduce Yourself forum in a bit.  But first, some math.

Quick Answer

Suppose there are n people with n different professions.  If one makes a random matching of the list of people to list of professions, then the probability that one correctly matches exactly k professions is

(nCk) (!(n-k))

Some explanation

This is actually a somewhat difficult problem, and I don't really know if there is a slicker way to do it.  What that means is I'll have to introduce a concept that I think is a bit esoteric.  (Actually, I just learned about the subfactorial trying to figure this problem out, and I'm a math grad student, so I've seen my fair share of such things.)

First, the conceptual definitions:

-- An integer n factorial (written n!) is the number of ways you can order n different objects.  For example, 3! = 6 because, if you have three objects a,b and c, you can order them 6 ways: abc, acb, bac, bca, cab, and cba.

-- An integer n subfactorial (written !n) is the number of ways you order n different objects so that none of the objects are in the right place, i.e. the first object isn't in the first place, etc.  For example, !3 = 2 because, of the 6 ways of ordering a, b and c above, only bca and cab don't have a in the first place, b in the second or c in the third.

-- An integer n choose k (written nCk) is the number of distinct subsets of size k you can make from n objects.  (Order doesn't matter).  For example, 3C2 = 3, because of {a,b,c}, you can make 3 subsets of size 2: {a,b}, {a,c} and {b,c}.  (Order doesn't matter, so {a,b} is the same as {b,a}.)

Second, the technical definitions:

-- The factorial of n the product of the consecutive series of whole numbers from 1 to n.  For example, "5!" is shorthand for "5*4*3*2*1" and "3!" is shorthand for "3*2*1".  So 5!=120 and 3!=6.     (0! = 1 = 1! by definition).

-- The subfactorial is defined as
!n = n! (1 - 1/1! + 1/2! - 1/3! + 1/4! - ... + 1/n!)
(Note that  1/3! should be read as 1/(3!) = 1/6, not (1/3)!.  Also, the sign in front of 1/n! is + if n is even, - if n is odd.)
So, as an example:
!5 = 5*4*3*2*1 [1 - 1/1 + 1/(2*1) - 1/(3*2*1) + 1/(4*3*2*1) - 1/(5*4*3*2*1)]
!5 = 120 [1 - 1/1 + 1/2 - 1/6 + 1/24 - 1/120]
!5 = 44

If you look up subfactorial, you can find the definition written more clearly.

-- n choose k is defined as
nCk = n! / [k! * (n-k)!]
So, as an example:
3C2 = 3!/[2! * (3-2)!]
3C2 = 3!/[2! * 1!]
3C2 = 6/[2*1]
3C2 = 6/2 = 3

Sketch of justification

Consider the 5 women (woman A, B, C, D and E) and 5 professions proposed in the original problem and say we wanted to know the probability that a guess gets exactly 2 correct.

For illustration, let's say the guess ABCDE is completely correct.  So an example in which we get exactly two correct is EBCAD.

First we need to figure out which 2 we get correct.  There are 5C2=10 different ways we can get 2 correct, because essentially we're just choosing 2 of the women and assigning the correct professions.  (This is where the nCk comes in to play.)  In our example, we get the professions correct for woman B and C, but we could have gotten it correct for 9 others: {A,B}, {A,C}, {D, E}, etc.

Now focus on one possible correct pair, say {B,C}, so our list looks like *BC**.  We need to fill in the other three women's professions and we need to get them wrong.  Otherwise, we'd have more than two correct.

If you think about it for a second, you're just trying to list A, D and E is an order so that A is not first, D is not second and E is not third.  But, by our conceptual definition of subfactorial, the number of ways you can do this is !3 = 2.  (There is where the !(n-k) comes into play.)

So, for each correct pair we can choose, there are !3=!(5-2) ways we can order the remaining three professions so that they are incorrect.  Thus there are
(5C2)*(!(5-2)) = 10*2 = 20
different ways we can guess the 5 professions and get exactly 2 correct.

Finally, to find the probability, we need to divide this by the total number of guesses we can make, which is 5!=120, because a guess is simply a specific ordering of the five letters.  (See the conceptual definition of factorial.)

So the probability we get exactly 2 correct is
------------------- = 20/120 = 16.7%.

I hope this explanation is clear enough for digestion.  Let me know if you have any questions.

Offline Chew

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Re: SGU 5x5 #99
« Reply #9 on: January 13, 2011, 10:18:55 PM »
It is very clear. It' a lot to digest but I will definitely digest it.

Thanks for all the work you put into that.

Again, welcome to the forum.
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Offline GodSlayer

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Re: SGU 5x5 #99
« Reply #10 on: January 16, 2011, 06:33:14 AM »
Is there a quick algorithm for computing the odds in situations like this? I've seen psychics tested like this before and I'd love to be able to calculate the probability distribution because the host always says the simple 1/n probability which, as you pointed out, is wrong.

isn't this why Randi spoke of the binary test for water dousing -- two pipes, one with water, one with none. 50% shot, if you can douse, beat those odds.

if someone claims to be able to tell profession, sex, marriage status, whatever, from hand-writing, why not say 'ok, is this one a man or a woman?' for a series. get a series where 50% are teachers and 50% are something else, and run a series of 'is this person a teacher?' -- as many yes or no tests of their ability as they claim to have insights of, and see if they beat a good old 50% reliably/consistently either on a given one of their supposed insights or across the board of their whole package of claims is true.

i haz h8 for math.
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