What’s the probability that in the next ten years, a nuclear bomb will be detonated on only the western half of the United States? We don’t know, so 50-50 is our best guess.

What’s the probability that in the next ten years, a nuclear bomb will be detonated on only the eastern half of the United States? We don’t know, so 50-50 is our best guess.

What’s the probability that no nuclear bomb will be detonated in the United States? We don’t know, so 50-50 is our best guess. And, of course, the probability of two bombs, one in the eastern half and one in the western is also 50-50.

So now we have four possible cases, all mutually exclusive, one of which must occur. And our best guess for the probability of each one is 50-50.

Similarly, our guess that at least one bomb will be detonated is also 50-50. So we have two events, each with a 50-50 chance. And the probability that both will occur is 50-50, the probability that neither will occur is 50-50, and the probability that one but not the other will occur is also 50-50.

Surely we can do better than these obviously nonsensical alleged “best guess”es.

]]>@Porlock: Evens and odds were on the table because besides having one-to-one correspondence, they also are equiprobable. That is, in the limit, as an interval of integers gets larger without bound, the fraction of evens and the fraction of odds both equal 1/2. That is, of course, not true of squares and non-squares.

]]>I’m not sure why it is that the statement is surprising. Zero and one both behave the same way – being squares of themselves. Two and all numbers beyond have corresponding squares.

Is it because of the behavior of zero and one that gives people trouble imagining this? It’s also not difficult to imagine if one is geometrically-minded: an actual square of any arbitrary unit in whole numbers can be constructed.

Zero and one are certainly naughty numbers, though. Paired up (for instance, in identity matrices in linear algebra) they wreak havoc. The proof of the PNT is concerned with the area between both numbers (the “critical strip”) in which all of the nontrivial zeroes of the Riemann zeta function are proven to reside. They’re also the only numbers one needs in order to represent any other number (the binary numbering system being the simplest).

As the “dynamic duo” of mathematics, I guess one should rather expect them to possess super powers. ðŸ˜‰

]]>Again, think: one to one correspondence. The proof is not difficult.

This has been called Galileo’s Paradox, a pretty fair eponym, since it appears in Two New Sciences (1638) and does not seem to have been stated with the same clarity by any earlier author.

]]>You have failed to prove, of course, that the chances that the LHC can destroy the world are 0 when it is turned off. It may be that an unpowered LHC contains more dangers than a powered LHC.

ðŸ™‚

In general, of course, people have forgotten to consider the post-black-hole-creation scenario: How do you comprehend “better”, “worse” or even “destroy” in an environment where you cross a black hole boundary?

If a black hole event does occur, maybe heaven and earth as we know it will be destroyed and be instantly replaced with a better one.

At another level, what if we’re crossing black hole boundaries all the time, but just don’t know it?

–recherche

]]>@Charles: I think it’s clear that Charles Murray does understand that about averages. But I also think that IQ is a pretty crude way to assess people’s intelligence. In my opinion, it measures the dot product between the test taker’s mind and the test maker’s mind, and little else. (I understand it’s somewhat well correlated with future salary, though.)

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