I am going to pull out some familiar chestnuts here. If you have not heard these puzzles before, then good, you may find them interesting. If you have heard them before, even better.
First, let’s start with something obvious. If you toss a fair coin once, does the result tell you anything about the result of the next toss? No.
Which leads to…
The Twin Paradox
(Yeah, I know that the name is already taken).
A mother pushing a pair of non-identical twins in a baby carriage calls you over. She tells you that one of them is a boy.
What is the probability that the other child is a boy?
And this is generally where the instructor gets to condemn everyone who answers 1/2.
Here are the possible sex groupings of the twins:
1. BB
2. BG
3. GB
4. GG
Only number 4 can be ruled out. Therefore the chance of there being two boys is 1/3, not 1/2.
The Game Show Paradox
On a game show there are three curtains. Behind two of the curtains are goats. Behind the other curtain is a Ferrari. The contestant has to choose one of the curtains to open, and he gets whatever is behind that curtain.
So Bob the contestant picks a curtain to open. Before it is opened, however, Pat the host opens another curtain, showing the goat behind it. He then gives Bob the opportunity to switch curtains. Should Bob switch?
And this is generally where the instructor gets to condemn anyone who says it doesn’t matter.
When Bob picked the first curtain, there was a 1/3 chance that the Ferrari was behind it, and a 2/3 chance that it was behind another curtain. So it is obvious that if he stays with the curtain he is at there will only be a 1/3 chance of winning the Ferrari. But if he switches, he will have a 2/3 chance.
No Good At Probability?
Nearly everyone’s gut feeling was 1/2 and ‘doesn’t matter’ respectively. This proves we’re no good at probability, right? Not so fast.
There is actually a good reason for our gut feelings. And it is the same reason in each case. Let’s go back to the twins.
Scenario A: Imagine that you had a great many mothers and their twins together in a room. You collect all of the mothers who could can truthfully say “one is a boy.” This is 3/4 of the mothers. Then you check the sex of either one of children and record your results.
Scenario B: Imagine instead the same situation, except this time you ask the mother to tell you the sex of either one of the children. When the mother says “boy” – true for 1/2 of the mothers – you then check the sex of either one of the children.
So here is the secret to the twin paradox: We solved the twin paradox as A, but B was how we heard it.
B is a narrative – in other words a sequence of events. That is how we tell stories. The twin paradox was presented as a narrative.
For want of a better term, A can be referred to as a description of a system. That is how we usually set up mathematics problems. We describe a system and then solve it. The twin paradox was solved in this manner.
The game show paradox is exactly the same.
Scenario A: Imagine a game show where the host knows where all of the prizes are. When a contestant picks one curtain, the game show host will always pick out one of the other curtains where a goat is.
Scenario B: Imagine a game show where the host knows nothing about the prizes. When a contestant picks one curtain, the game show host will pick a curtain at random and show what is behind it.
We solved the game show paradox as A, but B was how we heard it.
So this is not a problem of being good at probability or not. Our minds solve these problems just fine. It is a matter of language being too inexact to specify to our mind just which problem it should be working on.
Posted by Thrasymachus at 11:12 PM | | TrackBack Don’t sneeze @ genes
The Genetics of Allergies: Hereditary factors linked to more severe response to diesel fumes.
Posted by razib at 07:06 PM | | TrackBack It could be worse….
I guess I might be seen as acting a bit hysterical recently about the amnesty plan-but I lived in a compound until I was 4 and am not too excited about going back. But, in any case, some perspective-we don’t have it as bad as much of Europe. Case in point, Denmark (my comments in red):
One of the key mistaken assumptions made by various Danish governments was that after a generation, the children of the newcomers would assimilate, marry Danish girls, and become jolly Danes themselves. In fact, rather than marrying locally, most Turks, 95 percent in Rockwool’s reckoning, still import a Turkish wife even in the third generation. In fact, many Turks feel an obligation to help cousins back in the ancestral village get out through arranged marriages….
…
Because of the poor language abilities and work habits of many foreigners, employers are reluctant to hire them. As a result of unemployment, the father, the traditional seat of authority in Islamic families, often loses the respect of his children. The result has been a sharp increase in crime among second generation immigrants — to the extent that in the public mind, the word crime now brings to mind the image of a foreigner.
[Thank god that the United States tends to restrict immigration from Third World Muslim countries to skilled immigrants (more or less), otherwise I would have a hard time getting a date]
A rash of gang rapes over the past year has caused particular consternation. In one highly publicized case, seven Palestinian youths who were accused of gang-raping a teenage girl got off with extremely light sentences — three months — and were seen celebrating afterwards…
[In Denmark, where predominantly Muslim immigrants account for 68% of rapes, Islamic “community leaders” went out of their way to describe rape as “un-Islamic.”]
…
Finally, there are the financial costs. Immigration and the failure of integration have been staggeringly expensive, a tremendous strain on the welfare state. A minority of 4 percent of the population — that is, non-Western immigrants — accounts for fully 34 percent of the Danish social budget….
…
There have been some legislative efforts to address the problems. One new law seeks to bar immigrants under 25 years of age from bringing a foreign spouse into Denmark. This is done expressly to prevent arranged marriages: Older, more mature immigrants, it is believed, are less likely to give in to the dictates of family and custom….
[because of the EU common borders, many Muslims are moving to Sweden, importing their spouses from the mother country, and after a few years planning to move back to Denmark]
You give asylum to semi-literate rustics from Muslim countries and expect them to become good Danes in one generation!. Big surprise that they soon become alienated from a society they can’t succeed in, a society that assumes they just aren’t up to snuff because of past experience, and take refuge in pre-modern religious and cultural traditions. Groups like the Arab European League are a synthesis of Western narcissism and Eastern piety-the reactionary medieval traditions of their volk are a salve to their bruised egos.
Read the full article.
Posted by razib at 04:41 PM | | TrackBack Variation on Universal Grammar?
This article in The Economist highlights research that indicates that language affects the way we look at the world. Much of the same territory is tread by The Geography of Thought, and I suspect that the linguists will find that this is only an overlay over the “Universal Grammar,” just like the author of The Geography of Thought pretty much admitted by the end of the book that individuals could easily be re-trained to “think differently” (more Western or Eastern). Of course, language acquisition capacity drops a great deal after adolescence, so there are differences. The researcher mentioned above is an evolutionary psycholinguist, so I am skeptical that he will give much comfort to the tabula rasa crowd who are prone to assert that ideas precede the reality of the world around us.
Posted by razib at 03:55 PM | | TrackBack Dumb & dumber
Sometimes I wonder if the New Agey Left and the Medieval Right are going to collude to undermine the Modern world. Check out this moronic behavior:
Some Nigerian Muslims believe that the polio vaccine is part of an American plot to depopulate poor countries. Some imams preach that it is laced with anti-fertility drugs, can cause AIDS and is even linked to mad-cow disease. The source of these interesting theories was apparently American websites promoting “holistic” medicine.
Full article in The Economist (there isn’t that much more).
Reminds me of the idiocy of the pro-non-iodized salt movement in India.