I just watched the same video and it was one of the most insightful things that I have seen recently. Feynman is simply acknowledging that there are a few things in physics which are very difficult reduce to layman concepts. The other bit, about "why questions", is also quite interesting. I have come to believe that a lot of questions which sound very interesting, like "why do I exist?", don't really have good answers, and probably will never have. e.g. if it did turn out that we are living in a matrix, the question will simply reduce to "why does the matrix exist?" and soon. Thats what Feynman is trying to point out that "Why something happens?" is not a complete question unless we agree to some understanding beforehand. Otherwise you just go deep into the rabbit hole.
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This is pretty hilarious from the amazon review quoted in the article:
But even with these problems, The Story About Ping has earned a place on my bookshelf, right between Stevens' Advanced Programming in the Unix Environment, and my dog-eared copy of Dante's seminal work on MS Windows, Inferno. Who can read that passage on the Windows API ("Obscure, profound it was, and nebulous, So that by fixing on its depths my sight -- Nothing whatever I discerned therein."), without shaking their head with deep understanding. But I digress.
The OP is correct. Cardinality in mathematics is the number of elements in a set[1]. What you are probably referring to is the concept of cardinal numbers[2]. I guess you can skip the smug tone.
We should be glad that open source (Linux/Ubuntu) works when it does. Its such a curious case that its expected of Linux to support all myriad kinds of hardware systems. Its the responsibility of the hardware vendor to provide the drivers, not the hackers who labor hard to reverse engineer the specs when the vendors even refuses to publish how the hardware works. Few vendors really care about Linux. On the other hand, Windows has it really easy. Before the release of a new version, vendors work their asses off with Microsoft to ensure that before the OS reaches market, their drivers are already up to date. Apple has to care about only one piece of hardware which is manufactured by them (leaving peripherals of course).
We should be just glad that it works when it does. Those who work to make it work, work hard.
Disclaimer: On my Lenovo laptop, today, every single device works now on Ubuntu 9.04 from the card reader to webcam. It didn't always.
Lets just say that nobody reads popular science/physics books for "real" understanding. I always thought of them more as entertainers, giving a glimpse into the mysteries of modern physics, even if superficial or exaggerated, clearer than what probably the situation at the ground is. Not every one can spend enough time to go through the painstaking path of understanding physics up to the level of Quantum Mechanics or General Relativity. If I really wanted to understand what was happening, I would go pick up David Griffiths.
As you said, "average the overall probabilities between a regular game and a switch-style game", which is 1/3 for a game with no-switching and 2/3 for a game with always-switching. Averaging gives you a probability of 1/2. So if you switch, you win 50 % of the time. But so do you if you do not switch. So there is no benefit in switching, which was basically my point.
EDIT: In both the versions of the game, I am assuming an omniscience host who always opens the door with a goat behind, if he does open a door at all.
There is a certain subtlety to this problem which many people overlook. It is that whether the host <em>always</em> gives the option to switch. Only if the host always gives you the option to switch, there is a benefit in switching. If the host acts randomly(say deciding on a coin flip whether to give the option to switch), then the benefit of switching is nullified. Also, if the host is malicious (gives the option to switch only if you have the door with the prize), then of course switching decreases your chances of winning.
I believe that the reason several "PhDs" were confused over the answer has got to do with the fact that they might have assumed a host which acts randomly.