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And "with probability one" doesn't mean the same as "must ... and one, the event of picking 0.5 has probability zero.

Must... resist... getting... sucked... in... But can I ask what that "uniformly" is supposed to signify? I fail to see any use of random or randomly, so I wonder if this is supposed to modify the verb, as some sort of alternative to "randomly."

Sorry. "Pick real numbers randomly according to a uniform distribution" (i.e., one where all possible numbers have equal probability of being picked). I deal with random numbers a lot, and it's a common shorthand to just mention the distribution and assume that people know that it refers to the method of picking. It's an unwarranted assumption here.
Or does it modify "real," and so what does it mean to be uniformly real as opposed to just plain real? Or is it "nothing but" real numbers? And you're assuming "randomly" is part of the definition of "pick"?

Only when combined with the description of how to pick ("uniformly" with probabilities given by the uniform distribution").

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But what does "popularly" mean?

In this context, I meant outside mathematics. Was that not obvious?

My question was sorta rhetorical. I didn't really mean it to ask you what you meant by "popularly". Rather, I meant it as equivalent to something like, "But let's think about what "popularly" can really imply".
You say "outside mathematics", but is there really a sharp dividing line between inside and outside mathematics? Suppose an instructor is spoon-feeding elementary principles to a reluctantly attentive class and suppose the instructor in an attempt to instill at least a modicum of understanding makes a statement that is true except for an esoteric condition that would be tedious to discuss. If the instructor doesn't even mention the exception, is he or she inside or outside mathematics? Couldn't there be
practically a continuum of divergence from rigor?
(Lots of good stuff omitted.)
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And the probability of throwing a number smaller than three on a regular six-sided die?

You've out-subtled me again, John. I know there's a point there, but I don't see it.

Paul
In bocca al Lupo!
If you search the group for older maths threads, you ... am sure are not correct in any branch of maths.

But what does "popularly" mean? A response that seemed to be coming from a qualified scientist held that "exponential growth" is commonly understood to mean explosively rapid growth even among physicists who really know better.

I've often heard "exponential growth" used by people highly trained in mathematics, such as working physicists or chemists, to refer to explosive growth or growth exceeding other rates, so long as they are commenting in a way that doesn't constitute a full or proper mathematical description of what they are talking about or else are discussing a growth that will necessarily be monotone and the relevant point is whether it will grow faster than something else.

Of course one can fashion exponential arguments in a vast number of ways that are very different from the simple exponential rise in which growth is monotonic increase at a rate in proportion to quantity present. It seems the conspicuous feature of the simple exponential rise as exceeding other descriptions of growth is so commonly known that there has arisen a popular usage of "exponential" as referring to that particular conspicuous feature, despite the fact that this feature isn't invariably the case in exponential growth. One can applaud it as replacing misuse of "geometric progression" for activities that are not geometric progressions but could sensibly be described as exponentially growing, the exponential being so very versatile. Alternatively, one can say is simply introduces a new misconception and is no improvement.In this sense, a word used to refer to a conspicuous feature of a particular instance of its referent rather than all instances, a "popular" word is popular because of the popular misconception about it. In some quarters it might be popular to call the U.S. a "Christian nation", though Christianity has merely been a conspicuous feature of U.S. society and the phrase doesn't capture the secular nature of the country nor the inclusion of non-Christians from the start.

It might be popular to call the English a great sea-going people even if half of them in any given era might be tricked into agreeing that a "stern board" must be something primarily used in disciplinary matters on a ship for more recalcitrant sailors. Or "Hollywood" style might be popularly used to refer to something glamorous, of excessive display, or for any other number of conspicuous features which seem to ignore the whole of the movie business.
Aaron Davies turpitued: Careful! "With probability zero" does not mean the same as "can never happen".

And "with probability one" doesn't mean the same as "must happen". Somehow. If you pick (uniformly) real numbers between zero ... nearest I can come is that one "can't happen" while the other "can happen in any given trial, but won't".

I would rather say the probability of choosing 2 is undefined since it is not in the set of choices.
In the past times we've gone around on this, but I don't think I've seen anybody explain how one is ... merely the case that the expected (at any non-zero probability level) number of trials required to see them is infinite.

dg (domain=ccwebster)
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Aaron Davies turpitued: Careful! "With probability zero" does not mean the same as "can never happen".

And "with probability one" doesn't mean the same as "must happen". Somehow. If you pick (uniformly) real numbers between zero ... nearest I can come is that one "can't happen" while the other "can happen in any given trial, but won't".

Ah // threads, // worlds.
For an example, consider the question:
what is the probability of not throwing a six
in infinitely many throws.
The answer is of course zero,
but on the other hand it is obvious that it can happen, since all the throws are independent.
Best,
Jan
And "with probability one" doesn't mean the same as "must ... the other "can happen in any given trial, but won't".

Ah // threads, // worlds. For an example, consider the question: what is the probability of not throwing a six ... course zero, but on the other hand it is obvious that it can happen, since all the throws are independent.

Is it? One can construct a "probability space", modelling the tossing of a fair die infinitely many times, for which (i) the throws are independent (and in each throw the chance of tossing a 6 is 1/6) and (ii) it is impossible to not throw a 6 infinitely many times.

J.
If you search the group for older maths threads, you ... am sure are not correct in any branch of maths.

I started that thread, in 2003 I think. See the entry titled exponential growth at wikipedia.org . Mike Hardy
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And the probability of throwing a number smaller than three on a regular six-sided die?

You've out-subtled me again, John. I know there's a point there, but I don't see it.

Nought point three recurring is not a finite number.
John Dean
Oxford
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