Statisitcs - common fallacy

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Statisitcs - common fallacy

#1  Postby Shuggy » Jul 06, 2012 11:25 pm

I'm reading Thinking, fast and slow by Daniel Kahneman and it's excellent, but on p 115 he says:
Our predilection for causal thinking exposes us to serious mistakes in evaluating the randomness of truly random events. For an example, take the sex of six babies born in sequence at a hospital. The sequence of boys and girls is obviously random; the events are independent of each other, the number of boys and girls who were born in the hospital in the last few hours has no effect whatsoever on the sex of the next baby. Now con­sider three possible sequences:

BBBGGG
GGGGGG
BGBBGB

Are the sequences equally likely? The intuitive answer- "of course not!" -is false. Because the events are independent and because the outcomes B and G are (approximately) equally likely, then any possible sequence of six births is as likely as any other. Even now that you know this conclusion is true, it remains counterintuitive, because only the third sequence appears random. As expected, BGBBGB is judged much more likely than the other two se­quences.


I think he is confusing two popular fallacies with a statistical fact.

Fallacy 1. After six girls, a boy is more likely (or indeed, after 666 girls). This is obviously false.
Fallacy 2. BBGBGG is more random/more likely than GGGBBB or GBGBGB. This is less obviously false.

Fact. The sexes in a run of six are most likely to be 3G3B (in any order), then 2G4B or 2B4G, then 1G5B or 1B5G, with six girls or six boys the least likely. (The first child must be one sex - with one in 2000 intersexed, but we'll ignore that - it's one chance in two the next child will be the same sex, one in four there will be three in a row, one in 8 four in a row, and so on.) So in any run of six, the chances of all one sex are one in 2^5 or 32, and the chance of that sex being girls is half again, so the chance of GGGGGG is one in 2^6 or 64. Expressed differently, the chance of G is one in 2, of GG one in 4 (with equal chances of GB BG or BB) GGG one in 8 (vs GGB, GBB, GBG, BBB, BBG, BGG and BGB) etc.

So has Kahneman overstated his case, or have I missed something?
Last edited by Shuggy on Jul 06, 2012 11:43 pm, edited 1 time in total.
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Re: Statisitcs - common fallacy

#2  Postby Paul Almond » Jul 06, 2012 11:43 pm

Daniel Kahneman is correct. Each of the three sequences is equally likely.

However, if you grouped all the possible sequences into kinds of sequence, then some kinds of sequence would be more likely than others, because different kinds of sequences will correspond to different numbers of individual sequences.

e.g. How likely is it that all the children have the same gender (all B or all G)? There are only two individual sequences for which this is the case: BBBBBB and GGGGGG. On the other other hand, there are many more individual sequences for which it is not the case, so we can say that "all the same gender" is unlikely - but it would still be wrong to say that some particular sequence of the "all the same gender" kind (e.g. GGGGGG) is less likely than some particular sequence of the "not all the same gender" kind - which is the fallacy which he is discussing.
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Re: Statisitcs - common fallacy

#3  Postby igorfrankensteen » Jul 07, 2012 1:51 am

Kahnerman is indeed correct, he just is a lousy teacher. The fact that he confused YOU, proves that. His chosen way to explain his point is much more convoluted than it needs to be, and therefore is more confusing.

A better way to describe the situation, I think, would be to point out that there is no statistical difference between looking at INDIVIDUAL BIRTHS, and looking at SETS OF SIX INDIVIDUAL BIRTHS.
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Re: Statistics - common fallacy

#4  Postby Shuggy » Jul 07, 2012 4:29 am

Ah, I think I have it.
The chance of GGGGGG is one in 64.
The chance of BBBGGG is also one in 64.
The chance of BGBBGB is also one in 64.
There are 64 possible sequences, so "any possible sequence of six births is as likely as any other." as he says.

Since it is a book about how we think, I should look further into how I found that so hard. I seized on the rarity of GGGGGG, and clumped the mixed-sex birth orders (to my credit, only into different sex-ratio clumps, not one big clump), when the rarity of each of the birth-orders was what was to be contrasted with GGGGGGG, not the relative frequency of different ratios.

And thanks for the compliment, Igor.
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Re: Statisitcs - common fallacy

#5  Postby Microfarad » Jul 07, 2012 4:35 pm

Every sequence is equally probable.
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Re: Statisitcs - common fallacy

#6  Postby byofrcs » Jul 07, 2012 5:24 pm

Well other than the sex ratio not being exactly 50:50...
http://www.ncbi.nlm.nih.gov/pmc/article ... =pmcentrez
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Re: Statisitcs - common fallacy

#7  Postby Moridin » Jul 07, 2012 7:40 pm

byofrcs wrote:Well other than the sex ratio not being exactly 50:50...
http://www.ncbi.nlm.nih.gov/pmc/article ... =pmcentrez


The example clearly assumes equal probabilities of B and G.
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Re: Statisitcs - common fallacy

#8  Postby virphen » Jul 07, 2012 7:56 pm

And if you don't assume equal probabilities, Fallacy 1 is even more of a fallacy...

Fallacy 1. After six girls, a boy is more likely (or indeed, after 666 girls). This is obviously false.

After 666 girls in a row, you should be concluding not that a boy is more likely, but a girl is overwhelmingly more likely.
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Re: Statisitcs - common fallacy

#9  Postby Moridin » Jul 07, 2012 8:03 pm

virphen wrote:After 666 girls in a row, you should be concluding not that a boy is more likely, but a girl is overwhelmingly more likely.


No, because the assumption is that we are dealing with a fair coin. Or a fair uterus, as it were.
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Re: Statisitcs - common fallacy

#10  Postby virphen » Jul 07, 2012 8:07 pm

Moridin wrote:
virphen wrote:After 666 girls in a row, you should be concluding not that a boy is more likely, but a girl is overwhelmingly more likely.


No, because the assumption is that we are dealing with a fair coin. Or a fair uterus, as it were.


Please read my post more carefully.
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Re: Statisitcs - common fallacy

#11  Postby Shuggy » Jul 08, 2012 1:15 am

Moridin wrote: Or a fair uterus, as it were.
Don't blame the uterus. You would mean fair sperm if we were assuming fairness.
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Re: Statistics - common fallacy

#12  Postby Shuggy » Jul 08, 2012 1:20 am

Shuggy wrote:Ah, I think I have it.
The chance of GGGGGG is one in 64.
The chance of BBBGGG is also one in 64.
The chance of BGBBGB is also one in 64.
There are 64 possible sequences, so "any possible sequence of six births is as likely as any other." as he says.

I should have spotted this earlier, since I had already noticed the fallacy in another context: I have long scoffed at people who pay fortunes for personalised car licence plates, especially for "rare" ones like "MADONNA" and "JAG". ('You paid how much to remind people what brand of car they're screwed to?") All are equally rare - unique, in fact. That's the whole point of having licence plates.
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Re: Statisitcs - common fallacy

#13  Postby byofrcs » Jul 08, 2012 3:55 am

Shuggy wrote:
Moridin wrote: Or a fair uterus, as it were.
Don't blame the uterus. You would mean fair sperm if we were assuming fairness.


That's right - human sex is determined by the SRY gene found in the Y chromosome (thus from sperm). So we're default female unless this gene is expressed.
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Re: Statisitcs - common fallacy

#14  Postby Moridin » Jul 08, 2012 1:05 pm

I think you understand what I meant.
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Re: Statisitcs - common fallacy

#15  Postby Moridin » Jul 08, 2012 1:08 pm

byofrcs wrote:That's right - human sex is determined by the SRY gene found in the Y chromosome (thus from sperm). So we're default female unless this gene is expressed.


Actually, it is possible to have the SRY gene and still be phenotypically female. You cannot ignore the biological context.
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Re: Statisitcs - common fallacy

#16  Postby byofrcs » Jul 08, 2012 1:40 pm

Yes I didn't want to mention all the bizarre options. I read a lot about the SRY when arguing with Muslims as they believe that sex determination was decided 40 days after conception....e.g.
http://www.answering-christianity.com/m ... iracle.htm

It took a while to explain that the sex was determined at conception but expression would be visible at the 40 days if you looked at abortions. They seem to think the genetic stuff happens later but in reality the sex of the foetus is already determined at conception.

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Re: Statisitcs - common fallacy

#17  Postby Moridin » Jul 08, 2012 1:41 pm

The genotypic sex.
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Re: Statisitcs - common fallacy

#18  Postby susu.exp » Jul 08, 2012 3:08 pm

One possible way to define genotypic sex.
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Re: Statisitcs - common fallacy

#19  Postby Microfarad » Jul 08, 2012 4:01 pm

byofrcs wrote:Well other than the sex ratio not being exactly 50:50...
http://www.ncbi.nlm.nih.gov/pmc/article ... =pmcentrez

Negligible
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Re: Statisitcs - common fallacy

#20  Postby Moridin » Jul 24, 2012 9:09 am

susu.exp wrote:One possible way to define genotypic sex.


What I am saying is that SRY region determines genotypic sex, but from thereon, it is much more complex.
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