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What he proved is that given any number n, there exist prime numbers p and q that are (1) greater than n and (2) within 70 million of each other. For example, for a particular n, perhaps p = n + 1,000,000,000 and q = n + 1,069,000,000.
This is not quite the same as saying there's necessarily a prime between n and n + 70,000,000. In fact, since the average density of primes decreases as 1 / log(n), I'm pretty sure that statement is known to be false.
It seems like there are three possible laws:
1) No (legal) marriage at all.
2) Legal marriage for everyone.
3) Legal marriage only for straight couples.
If you prefer 1 to 2 and 2 to 3, I have no issue with that. But saying "I'm against gay marriage" is a misleading and confrontational way to phrase those preferences.
If you prefer 1 to 3 and 3 to 2, I think that's misguided. Would you support eliminating interracial marriage? I'd support raising taxes on rich people, but if I supported raising taxes only on rich black people that would just be racist.
No vaccine today uses any live virus.
If you're interested in this stuff, you might want to check out the blog advancednflstats.com. It focuses only on the NFL, though.
If you build a model (pretty much no matter how you build it) and compare it to what coaches actually do, you'll find that in general, coaches don't go for it on fourth down nearly as often as they should. It's a big difference -- on average, coaches cost their teams about 1.5 wins per season by kicking too often on fourth down, and this is in a league where 8 wins is average, 10 will make the playoffs, and 12 will get you a first round bye. And it wouldn't be hard for them to do a lot better than they do. Saying that there are a lot of variables is just their weak excuse for not believing in the math.
In real life, it makes some sense to be risk averse. Money has decreasing utility the more of it you have, so it's completely rational to refuse to bet your life savings even if there's a 51% chance you'll win. In a game like football, however, utility is necessarily proportional to the probability that you'll win the game. If a play increases that probability on average, it's the correct play no matter how risky it is. Football coaches are treating the probability of winning as if it had decreasing utility. They're being irrationally risk averse.
Once you start looking for irrational risk aversion in games, you start seeing it everywhere. It comes up just as often in Jeopardy as it does in football -- people tend to risk less than they should when they hit a daily double (early in the game, you should almost always bet everything). Funny enough, I don't think Watson got it right either -- in the second half of the first match, it made a relatively small bet when the rational strategy was probably to bet everything.
I typed up my own thoughts on the different definitions of the word "irony" and published them here.
According to wikiped...er, my own research, irony can't simply mean "something bad happens to a good person." Tragic irony, in particular, is a subset of dramatic irony, which requires that the audience knows something that some character doesn't know. That certainly doesn't apply to any of the situations in the Alanis song.
'unix' is a preprocessor constant:
#define unix 1
(It should be defined like that on unix/linux systems)
Ah, that's exactly what I was missing. Thanks!